(NASA-CR-4287) STATIC AND FREE-VIBRATIONAL

RESPONSEOF SEMI-CIRCULAR GRAPHITE-EPOXY

FRAMESWITH THIN-WALLED OPEN SECTIONS Final

Report (Virginia Polytechnic Inst. and

State Univ.) 216 p CSCL 20K H1/39

N90-ZTZ22

Unclas

0280223
NASA Contractor Report 4287

Static and Free-Vibrational

Response of Semi-Circular

Graphite-Epoxy Frames With

Thin-Walled Open Sections

j. Scott Collins and Eric R. Johnson

Virginia Polytechnic Institute & State University

Blacksburg, Virginia

Prepared for

Langley Research Center

under Grant NAG1-343

N/ A

National Aeronautics and

Space Administration

Offpce of Management

Scientific and Techntcal

Information DwJsion

1990
Static and Free-Vibrational Response of Semi-Circular

Graphite-Epoxy Frames with Thin-Walled Open Sections

by

J. Scott Collins

Dr. Eric R. Johnson, Chairman

Aerospace and Ocean Engineering

(ABSTRACT)

Experiments were conducted to measure the three-dimensional static and free

vibrational response of two graphite-epoxy, thin-walled, open section frames. The frames

are semi-circular with a radius of three feet, and one specimen has an I cross section and

the other has a channel cross section. The flexibility influence coefficients were measured

in static tests for loads applied at midspan with the ends of the specimens clamped.

Natural frequencies and modes were determined from vibrational tests for free and

clamped end conditions. The experimental data is used to evaluate a new finite element

which was developed specifically for the analysis of curved, thin-walled structures. The

formulation of the element is based on a Vlasov-type, thin-walled, curved beam theory.

The predictions from the finite element program generally correlated well with the

experimental data for the symmetric I-specimen. Discrepancies in some of the data were

found to be due to flexibility in the 'clamped' end conditions. With respect to the data

for the channel specimen, the correlation was less satisfactory. The finite element analysis predicted the out-of-plane response of the channel specimen reasonably well, but

large discrepancies occurred between the predicted in-plane response and the experimental data. The analysis predicted a much more compliant in-plane response then was

observed in the experiments.

Ir-_-_-_-_-_-_-_-__ j_/___ _,INTENT!ONALLY BLANI(

PRECEDING PAGE BLANK NOT FILMED

iii
Table of Contents

Introduction ............................................................ 1

Background .......................................................... 1

Objective ............................................................ 8

Literature Review ...................................................... 8

Dynamics .................................................... 9

Statics ...................................................... 12

t

Summary ........................................................... 14

Test Specimens ......................................................... 15

Design and Fabrication ................................................. 15

Specimen Characterization ............................................... 20

Experimental Apparatus and Tests ........................................... 28

Dynamic Tests ....................................................... 29

Free-Free Frame Tests .......................................... 29

Clamped-Clamped Frame Tests ................................... 31

Static Tests .......................................................... 37

Table of Contents iv
In-Plane Static Tests ........................................... 39

Out of Plane Static Tests ........................................ 44

Crush Test .......................................................... 49

Experimental Results .................................................... 52

Dynamic Test Data .................................................... 52

Static Test Data ...................................................... 61

Load-Displacement Plots ........................................ 63

Data Reduction ............................................... 71

Flexibility Influence Coeft]cients ................................... 72

Crush Test .......................................................... 78

Analysis .............................................................. 83

Thin-Walled Curved Beam Theory ......................................... 84

Strain Energy ................................................. 84

Kinetic Energy ................................................ 97

Hamilton's Principle ........................................... 100

Hooke's Law ................................................ 102

Continuum Solutions ................................................. 110

In-Plane Vibration Solution ..................................... 111

In-Plane Static Solution ........................................ 115

Out-of-Plane Vibration Solution .................................. 120

Finite Element Computer Program ........................................ 124

Element Formulation .......................................... 125

Application to Test Specimens ................................... 127

Correlation of Numerical and Experimental Results .............................. 134

Table of Contents v
Correlationof DynamicData ............................................ 134

Free-FreeData .............................................. 135

Clamped-ClampedTests ........................................ 136

Correlationof StaticData .............................................. 151

I-SpecimenResults ............................................ 155

ChannelSpecimenResults ...................................... 159

Concluding Remarks .................................................... 161

Future Work ........................................................ 164

REFERENCES ....................................................... 166

Vibrational Test Data ................................................... 169

Frequency Response Data from Dynamic Tests .............................. 169

Mode Shapes ....................................................... 184

Static Test Data ....................................................... 193

Static Test Data Reduction ............................................. 193

Reduced Static Data .................................................. 197

Cross-Sectional Properties ................................................ 203

Table of Contents vi
List of Illustrations

Figure

Figure

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Figure

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Figure

Figure

Figure

I. Drop test of full scale fuselage section .......................... 3

2. Drop test of full scale fuselage section .......................... 4

3. Drop test of a composite fuselage subsection ..................... 6

4. Drop test of a composite fuselage subsection ..................... 7

5. Cross sections &the specimens .............................. 16

6. Specimen layup .......................................... 18

7. Specimen dimensions ..................................... 19

8. Ultrasound scan of the channel specimen ....................... 21

9. Three-point bend test configuration ........................... 22

10. Load-displacement data from three-point bend tests ............... 23

11. Iosipescu shear test results ................................. 25

12. Tension test results ....................................... 26

13. Free-free frame vibrational test setup .......................... 30

14. Structural analyzer test flowchart ............................ 32

15. Experimental apparatus for clamped-clamped dynamic tests ......... 34

16. Experimental apparatus for clamped-clamped dynamic tests ......... 35

17. Experimental apparatus for clamped-clamped dynamic tests ......... 36

18. Frequency response plots for the I-section specimen .............. 38

19. Load fixture and Coordinate system .......................... 40

List of Illustrations vii
Figure

Figure

Figure

Figure

Figure

Figure

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Figure

Figure

Figure

Figure

Figure

Figure

Figure

20. Load fixture ............................................ 42

21. In-plane test apparatus .................................... 43

22, Schematic of modified load fixture ............................ 45

23. Photograph of modified load fixture and dial indicators ............ 47

24. Out of plane test apparatus ................................. 48

25. Crush test apparatus ...................................... 50

26. First free-free in-plane mode of the channel specimen .............. 54

27. Second free-free out-of-plane mode of the I-specimen ............. 55

28. First free-free torsional mode of the I-specimen .................. 57

29. Comparison of free-free natural frequencies ..................... 59

30. Comparison of clamped-clamped natural frequencies .............. 62

31. Radial load versus radial displacement ......................... 64

32. Radial load versus lateral displacement ........................ 65

33. Radial load versus twist displacement ......................... 66

34. Lateral load versus radial displacement ........................ 68

35. Lateral load versus lateral displacement ........................ 69

36. Lateral load versus twist displacement ......................... 70

37. Crush test load displacement plot ............................ 80

38. Initial failures for crush test ................................ 82

39. Cylindrical and cross-sectional coordinates ..................... 85

40. General symmetric cross section ............................. 88

41. Loading for in-plane static solution .......................... 118

42, Strain energy distributions in vibrational modes for both specimens .. 129

43. Number of elements convergence study ....................... 130

44. Penalty parameter convergence study ........................ 132

45. Free-free in-plane mode shapes for the I-specimen ............... 137

List of Illustrations viii
Figure

Figure

Figure

Figure

Figure

Figure

Figure

46. Clamped-clamped in-plane natural frequencies for the I-specimen .... 139

47. Effects of torsional spring end conditions ...................... 142

48. Clamped-clamped, in-plane natural frequencies for the channel specimen 143

49. Five noded, in-plane, clamped-clamped mode shape for both specimens 145

50. Six noded, in-plane, clamped-clamped mode shape for both specimens 146

51. Clamped-clamped out-of-plane natural frequencies for the I-specimen 147

52. Clamped-clamped out-of-plane natural frequencies for the channel

specimen .............................................. 149

Figure 53. Five noded, out-of-plane, clamped-clamped mode shape for both specimens ................................................. 152

Figure 54.

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

55.

56. Proximity probe orientations tbr clamped-clamped vibrational tests

57. I-specimen frequency response plot for radial excitation; 11-111 ttz

58. I-specimen frequency response plot for radial excatation; 110-180 Hz

59. I-specimen frequency response plot for radial excitation; 233-400 Hz

60. I-specimen frequency response plot for lateral excitation; 6.8-15 Hz

61. I-specimen frequency response plot for lateral excitation; 20-260 I-Iz

62. Channel frequency response plot for radial excitatmn; 30-160 Hz

63. Channel frequency response plot for radial excitatmn; 142-240 Hz

64. Channel frequency response plot for radial excitation; 270-340 Hz

65. Channel frequency response plot for lateral excitation; 6.8-10 Hz

66. Channel frequency response plot for lateral excitation; 15-80 Hz

67. Channel frequency response plot for lateral excitation; 70-125 Hz .... 181

68. Channel frequency response plot for lateral excitation; 140-219 Hz ... 182

69. Channel frequency response plot for lateral excitation; 230-330 Hz ... 183

Six noded, out-of-plane, clamped-clamped mode shape for both specimens ................................................. 153

Influence of torsional spring stiffness on ..................... 157

170

171

172

173

174

175

.... 176

... 177

... 178

.... 179

.... 180

List of Illustrations ix
Figure 70. Two and three noded in-plane, free-free mode shapes ............. 185

Figure 71. Three noded in-plane, clamped-clamped modes for both specimens . . . 186

Figure 72. Four noded in-plane, clamped-clamped modes for both specimens . . . 187

Figure 73. Seven noded in-plane, clamped-clamped modes for the I-specimen . .. 188

Figure 74. Two nodcd out-of-plane, clamped-clamped modes for both specimens 189

Figure 75. Three noded out-of-plane, clamped-clamped modes for both specimens 190

Figure 76. Four noded out-of-plane, clamped-clamped modes for both specimens 191

Figure 77. Seven noded out-of-plane, clamped-clamped mode for the I-specimen . 192

Figure 78. Schematic of rectangular block, coordinate system and dial indicators 194

Figure 79. Deformed cross section ................................... 195

List of Illustrations x
List of Tables

Table I. Nominal material properties for AS4/5208 from Narmco ............ 17

Table 2. Engineering constants from quasi-isotropic coupon tests ............ 27

Table 3. Natural frequencies and node locations of the free-free modes ........ 58

Table 4. Natural frequencies and node locations of the clamped-clamped modes . 60

Table 5. Flexibility coefficients for the I-specimen from tile in-plane tests ...... 73

Table 6. Flexibility influence coefficients for the channel specimen from the inplane tests .............................................. 75

Table 7. Flexibility coefficients for the I-specimen from the out-of-plane tests . . . 77

Table 8. Flexibility influence coefficients for tile channel specimen from the outof-plane tests ............................................ 79

Table 9. Correlation of the free-free in-plane natural frequencies of the I-specimen 136

Table 10. Correlation of clamped-clamped in-plane natural frequencies for the Ispecimen .............................................. 140

Table 11. Effects of boundary spring stiffness on the in-plane vibrational response

of the I-specimen ........................................ 141

Table 12. Correlation of clamped-clamped, in-plane natural frequencies for the

channel specimen ........................................ 144

Table 13. Correlation of clamped-clamped, out-of-plane natural frequencies for

both specimens .......................................... 150

Table 14. Correlation of flexibility coefficient for the I-specimen .......... 155

Table 15. Correlation ofthe flexibility coefficients for the I-specimen ......... 158

Table 16. Correlation of the flexibility coefficients for the channel specimen .... 160

List of Tables xi
Table 17. Static data from in-plane tests for 20 pound radial load ............ 198

Table 18. Static data from in-plane tests for 40 pound radial load ............ 199

Table 19. Static data from in-plane tests for 60 pound radial load ............ 200

Table 20. Static data from out-of-plane tests ........................... 201

Table 21. Static data from out-of-plane tests ........................... 202

Table 22. Cross-sectional Properties of both test specimens ................ 203

List of Tables xii
Chapter 1

Introduction

Background

Fiber reinforced composite materials are used increasingly in the primary structural

components of military aircraft. Use of composites in civilian transport aircraft is generally limited to secondary structural componcnts. However, the well documented advantages of composites are motivating designers to use composites more extensively.

Using composites in primary structural components of civilian aircraft raises the issue

of crashworthiness. Research has revealed deficiencies in the energy absorbing capabilities of fiber reinforced composite materials. Though composites are poor energy

absorbers on a material level, it may be possible to design composite structures which

absorb energy efficiently. Thus, the challenge to the engineering community is to design

composite structural components which carry the flight loads and perform satisfactorily

in a crash situation.

Introduction l
Crashworthiness is a complicated issue, but arguably the primary concerns in a

crash are to maintain a protective shell around the passengers and to keep decelerations

at a survivable level. Aluminum fails in a ductile manner, thus it absorbs energy when

it yields and fails. Fiber reinforced composites tend to fail in a brittle fashion which inhibits their ability to absorb energy. It has been found under certain special circumstances (e.g., tubes in axial compression) composite structures are able to efficiently

absorb energy via a stable axial crush (Ref. 1). Though crushable composites are an

encouraging development, application of this technology to aircraft design is not imminent.

Before further discussion of the crashworthiness of composite materials, it is informative to review some of the research that has been conducted on the impact response of aluminum structural components. Drop tests of full scale aluminum transport

fuselage sections have been conducted at the NASA Langley Research Center (Refs. 2,

3, and 4). The results of drop tests of two fuselage sections are shown in Figs. 1 and 2.

The fuselage section which includes the wing spar (Fig. 1) shows no structural damage,

however, very high inertial loads were transferred to the passenger level. The fuselage

section without the wing spar (Fig. 2) suffered extensive structural damage resulting in

lower inertial loads at the passenger level. Extensive bending failures of the fuselage

frames helped reduce the inertial loads transmitted to the passenger level indicating that

the fuselage frame is an important component in the impact response of a conventional

aircraft.

If aluminum frames play a significant role in energy absorption, then composite

frames may have to behave similarly to obtain acceptable energy absorption. Tests

comparing the impact response of composite and aluminum frames (Ref. 5) showed that

the frames fail in distinctly different fashions. The aluminum frames formed plastic

Introduction 2
ORIGINAL PAGE

BLACK AND WHITE PHOTOGRAPN

Figure 1. Drop test of full scale fuselage section: Fuselage section including the wing spars suffers

little structural damage during drop test, resulting in high inertial loads at the passenger

level. The section was dropped from a height of six feet resulting in an impact velocity of

twenty feet per second.

Introduction 3
r

ZO

2C._

ZC _

0 r_

0

I

Figure 2. Drop test of full scale fuselage section: Fuselage sections located fore and aft of the wing suffered extensive structural damage, resulting

in lower acceleratior_s at the passenger level. The section was dropped from a height of six feet resulting in an impact velocity of twenty

feet per second.
hinges at several locations around the frame without fracturing, thus maintaining structural integrity. Failure of the composite frames involved a complete fracture of the cross

section near the impact point. Complete fracturing of the frame is an undesirable failure

since loss of structural integrity implies an uncrashworthy design.

Tests of built-up composite fuselage subsections (Refi 6) further demonstrated the

tendency of composite frames to fail in a brittle manner. The photographs in Figs. 3 and

4 show a skeleton subsection before and after the drop test, respectively. The frames

of the subsection suffered numerous localized brittle fractures resulting in loss of structural integrity and minimal energy absorption. A drop test of another subsection which

had a layer of skin attached to the outside of the frames resulted in a single brittle fracture of each frame at the impact point. The skin helped the subsection maintain structural integrity. However, in both cases, very high inertial pulses were transmitted to the

'passenger level'.

These test results indicate that conventional designs for composite fuselage frames

fail to provide satisfactory impact behavior. The structural design of a composite frame

must complement the material system such that crashworthiness is achieved. Since unconventional designs are necessary to utilize composites while maintaining

crashworthiness, it is important to develop reliable analytical tools which can predict the

response of these designs. The reliability of the analyses should be established by comparisons with experimental data. This research attempts to provide some of this data and

to evaluate a new finite element computer code. The finite element evaluated here is expected to be incorporated in the crash analysis finite element program DYCAST.

Introduction 5
I-'

C),

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C; F_

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Figure 3. Drop test of a composite fuselage subsection: The photograph shows an all composite fuselage skeleton subsection prior to drop test.
• 8, , • • •

t_

I'-

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173:70

Iz

"q r--

I-r3

Orri

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Figure 4. Drop test of a composite fuselage subsection: Subsection was dropped such that the impact velocity was twenty feet per second. The

"-_ photograph of the impacted subsection shows numerous localized brittle failures.
Objective

]'he objectives of this research are to measure the three-dimensional static and vibratory response of two graphite-epoxy frame specimens, and to correlate the experimentally measured response with predictions from a computer program which uses a

newly developed finite element. ]'he test specimens are thin-walled, open section, semicircular frames with a nominal radius of three feet. One specimen has an I cross section

and the other has an asynunetric channel cross section. The finite element was recently

developcd by Noor et al. (Ref, 7), for the analysis of curved thin-walled bcams with open

sections. "/he flexibility influence coelticients were measured in static tests lor loads applied at the midspan with the ends of the frames clamped in supports. Dynamic tests

provided the natural frequencies and mode shapes of the frames for free-free and

clamped-clamped end conditions. It is important to validate the analysis for the linear

responsc of these frames before attempting the nonlinear analysis of frame collapse.

Literature Review

The scope of this research includes both the static and dynamic response of semicircular frames. Since few researchers choose to address both topics in a single document, it is convenient to divide the literature review accordingly.

Introduction 8
Dynamics

The vibrations of curved beams and rings is an old and recurring topic in the literature. In general, the vibratory response of a curved beam or ring encompasses in-plane

modes (deformations within the plane of curvature) and out-of-plane modes (deformations out of the plane of curvature). If the curved beam has a plane of symmetry coinciding with the plane of curvature, then the in-plane and out-of-plane modes decouple

and can be treated separately. Many early researchers took advantage of this. Only

relatively recently have researchers addressed the coupled vibrational problem of a

curved beam with an asymmetric cross section.

The first published paper addressing the dynamic response of a circular ring was

written by Hoppe (Ref. 8) in 1871. In 1888, Lamb (Ref. 9) determined the in-plane natural frequencies for a shallow ring segment with free-free end conditions. In 1892, Love

(Ref. 10) generated the differential equations governing the vibration of a complete ring,

solved the equations approximately, and presented frequency equations for the in-plane

and out-of-plane modes of a complete ring.

In 1928, Den Hartog (Ref. 11) used a Rayleigh solution to solve for the first inplane natural frequency for ring segments with clamped and pinned boundary conditions. Den Hartog found that if the arc length of the ring segment was sufficiently small,

the fundamental in-plane mode was extensional in nature possessing only two nodes

similar to the fundamental mode of a straight beam. Generally, if the opening angle was

greater than 80 °, then the fundamental mode had three nodes and was tlexural. For a

semi-circular ring segment with clamped boundary conditions, Den Hartog's solution for

the frequency co in Hertz reduces to

Introduction 9
0"6971 ( Elyy)_ (I 1) co -- a 2 Y

where EIr_ is the in-plane bending stiffness, _ is the mass per unit length, and a is the

radius of the ring segment.

In 1934, Brown (Ref. 12) used a modified Rayleigh solution to obtain the natural

frequency of the fundamental out-of-plane mode for a clamped circular ring segment.

For a semi-circular ring segment, Brown's solution for the frequency co in Hertz reduces

to

.10842 ( --C )1 (1.2)

a

assuming the ratio of in-plane bending stiffness to torsional stiffness is large

(k = EI_/C >_ 75) where C is the torsional stiffness as defined by Timoshenko (Ref.

13), ? is the mass per unit length, and a is the radius of the ring segment.

The vibrational response of curved beams and rings attracted moderate attention

during the 30's, 40"s, and 50's as researchers sought to improve their predictions by incorporating refinements such as rotatory inertia and transverse shear deformation. In

the interest of brevity, the body of literature from this pcriod will not be addressed as it

is of limited interest here. The interested reader may refer to two papers by Lang (Refs.

14 and 15) and the dissertation of Maddox (Ref. 16) for a complete review of this period.

In 1963, Hammoud and Archer (Ref. 17) published an interesting paper addressing

the coupled vibrational response of complete rings and ring segments with asymmetric

cross sections. They presented the coupled differential equations governing the in-plane

and out-of-plane vibrations and used an approximate solution to the decoupled problem

as their first guess in an iterative procedure to solve for the coupled natural frequencies.

Introduction 10
Hammoud and Archer limited their scope to solid cross sections as their solution does

not account for the effects of cross-sectional warping.

Maddox (Ref. 16) addressed the decoupled vibrations of a complete thin-walled ring

with one plane of symmetry. His solution accounts for transverse shear deformation,

rotatory inertia, cross-sectional warping, and shear center eccentricity. Maddox reduced

a higher order shell theory to a ring theory by integrating around the contour of the

cross section and making explicit assumptions on the shell displacements. Maddox

showed that shear center eccentricity and warping have significant effects on the response of some thin-walled rings.

Endo (Ref. 18) solved for the vibrational response of a complete ring with an arbitrary cross section. Kirkhope, Bell, and Ohnstead (Ref. 19) also solved for the coupled

vibrational response of a complete ring with an arbitrary cross section and presented

experimental data in order to test their solution. These solutions are restricted to solid

cross sections since warping is not included. Williams (Ref. 20) developed the differential equations for complete thin-walled rings and applied them to several examples including ring segments with symmetric cross sections. Williams included the effects of

transverse shear, rotatory inertia, and warping in his solution.

Rao (Ref. 21) solved for the out-of-plane vibrational response of complete rings and

ring segments. Rao included transverse shear deformation and rotatory inertia in his

solution. He found that he could model a variety of unusual boundary conditions.

Culver (Ref. 22) used Vlasov's (Ref. 23) thin-walled beam theory in his solution for the

decoupled out-of-plane response of a curved beam. Culver assumed the cross section of

the beam was doubly symmetric and treated the inertia terms as a distributed load in

Vlasov's static formulation. Culver does not consider the inertia terms associated with

warping.

Introduction 11
Gardner and Bert (Ref. 24) presented a new first approximation theory for the inplane dynamic behavior of shear deformable structures. Several references are made

about the application of this theory to composite structures, but it was not addressed

explicitly. The paper does include some experimental data on thick full rings. Bhimaraddi

(Ref. 25) specifically addresses the dynamic response of curved laminated beams, though

his discussions are limited to rectangular cross sections. Bhimaraddi found that coupling

of in-plane and out-of-plane modes can happen despite a geometrically symmetric cross

section if the ring is an unsynm_etric laminate. Bhimaraddi presents the coupled

equations in terms of laminate properties and some experimental data from laminated

rings with two lamina.

With tile exception of Bhimaraddi (Ref. 25), the author knows of no research directed at the vibrational response of curved composite beams. Further, there seems to

be a dramatic lack of experimental data for curved beams of any cross section or material system. This research will help to fill these voids.

Statics

The static portion of the literature review will be limited to those papers which

specifically address thin-walled curved beams or have an important bearing on this research. Though much work preceded Timoshenko's series of papers (Rcf. 13) on thinwalled beams, his work unitied the existing engineering theories on the bending, torsion,

and buckling of beams with thin-walled open cross sections. He presented new developments such as the concepts of warping, shear center, and torsion center. His papers address thin-walled beams in a very thorough manner, though he does not treat curved

beams.

Introduction 12
Vlasov (Ref. 23) derived a beam theory for thin-walled beams including the effects

of curvature. This represents the first detailed discussion of curved thin-walled beams.

Vlasov included the effects of warping and rotatory inertia. He restricted his attention

to thin rings facilitating the neglect of transverse shear deformation and shear center

eccentricities. He also assumed that the cross section is infinitely rigid in its own plane.

Gjelsvik (Ref. 26) presents Vlasov's theory in more modern terminology though it does

not include the material on curved beams.

Bauld and Tzeng (Ref. 27) extended Vlasov's theory to account for anisotropic

material properties common in laminated composites. This theory uses modulus

weighted section properties to account for stiffness variations through the thickness of

the walls. This theory assumes that all the branches are midplane syrmnetric and the

beams are straight. Though the theory was developed for straight beams, it could be

extended to include curved beams. Lo (Ref. 28) used this theory in his analysis of

flexural-torsional buckling of laminated composite columns.

Mabson (Ref. 29) discusses some of the differences between isotropic and composite

curved thin-walled beams. Mabson shows that composite cross sections are more susceptible to flexural deformations than isotropic cross sections. This indicates that composite beams may be more likely to violate the assumption that the cross section is

infinitely rigid within its own plane. Mabson suggests a way of calculating effective

cross-sectional stiffnesses for composite laminated beams of various cross sections.

Tralli (Ref. 30) and Noor (Ref. 7) have both used hybrid finite elements to evaluate

the static response of thin-walled structures. Tralli's approach addresses only straight

beams, but will handle closed as well as open sections. While Tralli's program is limited

to static analysis, Noor's finite element was also developed for the free-vibrational

analysis of curved thin-walled beams. Noor's program is used extensively in this research and will be discussed more thoroughly in Chapter 5.

Introduction 13
Summary

Havingattemptedto providesomeof thebackgroundandmotivation tor this study,

the remainingmaterial will be devotedto describingthe research.The test specimen's

manufacture,material system,and dimensionsare discussedin Chapter2. The experimental apparatusand test proceduresare presentedin Chapter 3. The experimental

resultsarepresentedin Chapter4. The analyticaleffort is outlinedin Chapter5. Correlationof experimentalandnumericalresultscompriseChapter6. Concludingremarks

andrecommendationsfor futurework arepresentedin Chapter7.

Introduction 14
Chapter 2

Test Specimens

Design and Fabrication

Conventional aircraft designs use thin-walled frames and beams to build the skeleton of a fuselage. The test specimens for this research are intended to represent fuselage

frames. Two semi-circular frames were tested, a symmetric I-section and an asymmetric

channel section. The cross-sectional shapes are shown in Fig. 5. Channel and I-sections

are commonly used in conventional aircraft designs and provide the opportunity to investigate the effect of symmetry. In tests of composite fuselage subsections (Ref. 6), it

was found that the skin substantially strengthened and stabilized the subsections. This

motivated the addition of a thick layer of 'skin" to the outside of each frame. The layer

of 'skin' is intended to simulate the out-of-plane stabilizing effect of the fuselage skin.

The test specimens were fabricated using AS4/5208, graphite/epoxy,

preimpregnated, unidirectional, tape. The tape was manufactured by Narmco and the

specimens were laid up by Bell Textron. The nominal material properties for AS4/5208

Test Specimens 15
/

120

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0

;X

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0 r,_

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. ."-::::_.7,s.,-"..:,.i: :.!::::::::::::::::::::::: ===================== :-:.:i::(,. ::::;,,, :.

Figure 5. Cross sections of the test specimens: The photograph shows file cross sections of the l-seclion and channel section specimens. A sixteen

ply "skin' is bondcd to the outside flanges of each specimen. The cap flanges are the smaller flanges furthest from the skin.
are presented in Table 1. The specimens were laid up using a quasi-isotropic sequence

following the schematic in Fig. 6. The specified angles are measured with respect to the

circumferential axis of the specimen. The layup in the 8 ply cap flanges is (+45/0/90)_.

The layup for the 16 ply skin is (_+45/0/90)2_. In the region where the attachment flanges

are bonded to the skin, the 24 ply layup is

(+_451019019010/-T-45/++_45/O/90/+_45/0/90/90/O/-T45/90/O/T45).r. Although this layup is

asyrmnetric, a classical analysis of this region shows that the coupling terms are negligible. The web of the channel specimen has the same layup as the cap flanges, but the web

of the 1-specimen has an anti-synunetric layup (+_45/0/90/90/0/+_45).r, resulting in significant extension-bending coupling terms. It will be evident from the data in subsequent

chapters, that the anti-symmetric layup in the web of the I-section specimen did not

adversely effect its symmetric response.

Table 1. Nominal material propertiesfor AS4/5208 fi'omNarmco

Property

Longitudinal Modulus Ell

Transverse Modulus E2a

Trm_sverse Modulus E33

In-plane Shear Modulus G,=

Transverse Shear Modulus Gl3

Transverse Shear Modulus G23

In-plane Poisson Ratio vt2

Transverse Poisson Ratio v,3

Transverse Poisson Ratio v_3

I,amina Thiclmess t

Material Mass Density p

Value

18.40 x 106 psi

1.64x 10_ psi

1.64 x 106 psi

0.87 x 106 psi

0.87 x 106 psi

0.49 x 106 psi

0.30

0.30

0.35

0.0055 in

1.80 x 10-3 (lb-sec_)/(ft-in 3)

The nominal dimensions of the test specimens are shown in Fig. 7. The diameter

of the frames is 72 inches and the cross-sectional heights are about 1.5 inches. This geometry places the frames in the category of thin rings. The effects of rotatory inertia and

transverse shear typically have little effect on the response of thin rings.

Test Specimens 17
46

0

W

0

0

45

0

W

0

0

45

,t5 ,_5

45 -45

0 0

QO gO

9O 9O

0 0

-45 45

Skin

4S

0

45

45

0

0

-45

45

gO

0

45

45

Figure 6. Specimen layup: The layup for the channel section and I-section specimens is shown schematically.
F-, 5o,,_ F°7_'_

L k _ 1_

_J mJ A ,l

_1 =0.08" t=O.12"

I_ _= 2.50" _ _! I_ I_-_ 1.25" _"4 -! t t

,- 3.50" - _ 2"

°l

Figure 7. Specimen dimensions: The radial and cross-sectional dimensions 01"the tests specimens are

presented in two views.

Test Specimens 19
Specimen Characterization

Tile results of an ultrasonic inspection of the web of tile channel specimen is shown

in Fig. 8. ]'he white patches indicate regions which may be damaged or substandard,

raising questions about the consistency of the stiffness properties with respect to circumference. Subsequent tests were designed to investigate this concern. After the static

and dynamic tests were completed, the I-section specimen was instrumented for a crush

test and the channel specimen was cut into five ring segments for further evaluation.

Four of the segments were tested in three point bending to determine if the bending

stiffness was constant with respect to circumference. These specimens had a uniform arc

of about forty degrees. The fifth ring segment was cut into tensile and shear coupons for

material characterization tests.

The three point bend test configuration is shown in Fig. 9. The ring segments were

supported at their ends across a span of eighteen inches. The segments were loaded

radially by hanging twenty pound calibration weights from the center of the segments.

A load fixture was used to apply tile load to the segments such that the segments did

not twist. The displacements were measured using dial indicators. Each specimen was

tested five times and the resulting data was averaged. The results of the tests are presented in Fig. 10. The load displacement curves for the four segments are very consistent. The slope of the lines fall within _+3 % of the average slope. These results negated

much of the concern about circumferential stiffness nonuniformity suggested by the

ultrasound scan.

The last segment was cut into two tensile and three shear coupons for material

characterization tests. The coupon tests were intended to confirm the effective moduli

Test Specimens 20
ORIGINAL PAGE

BLACK AND WHITE PHOIOGRAPN

ORIGINAL PAGE IS

OF pOOR QUALITY

21
I:D

F"

>o

:7 o

U.3 ;_3

::K.Z

_mr-

.x__ -0

0

¢)

T.o

_;.

Figure 9. Three-point bend test configuration: Tile photograph shows the test apparatus ['or the three point bend tests. The ring segments were

supported at the ends and loaded at the midspan. The deflections were measured using two dial indicators.
8O

.0! .02 .03 .04

RADIAL DISPLACBAENT INCHES

SPECIMEN I _-e _ 2

SPECII_EN 3 _ SPEQMEN 4

.05

Figure 10. Load-displacement data from three-point bend tests: The plot shows the load versus displacement plot for the four segments cut from the channel specimen. The slopes of the lines

are very consistent indicating consistent bending stiffnesses.

Test Specimens 23
(E_, and G_) calculated from classical lamination theory and nominal material properties.

The shear coupons were cut from the web of the section since those coupons had

to be flat. The Iosipescu shear test was chosen because of the size limitations on the

shear coupons. 1"he shear coupons were cut according to the dimensions specified by

Pindera, et al. (Ref. 31). The coupons were instrumented with a three arm rectangular

rosette (electrical resistance foil gages) which had a gage length of two millimeters. The

results of the shear tests are presented in Fig. 11. The tests gave consistent values for the

shear modulus, but the coupons did not fail in shear because of the quasi-isotropic

layup. The laminates failed by edge brooming in the 90 ° layers at the contact points.

Two tension coupons, six inches by 5/8 inches, were cut from the cap flange of the

channel specimen. The initial geometry is curved, but the eight ply quasi-isotropic layup

is sufficiently flexible that the initial curvature has little effect on the test. The coupons

were instrumented with a three arm rectangular rosette on one side and a uniaxial gage

on the other side for bending correction. The two tension tests gave nearly identical results for the extensional modulus E_, and failure strain. The stress versus strain plots for

the two tensile tests are shown in Fig. 12. The failure strain in the tension tests was

about 1% for both coupons.

The results of the material characterization tests are summarized in Table 2. The

experimental values for E_ and G_ compared well with the analytical predictions from

classical lamination theory. The discrepancy in the extensional modulus was less than 1

%. The discrepancy in the shear modulus was about 8 %. The experimentally determined stiffness properties were used in all of the analyses for both the channel section

and the I-section specimens.

Test Specimens 24
25

2O

10

0

..... ! ...... ,. ! ...... ,., ! ........ ! ........

o.o o.2 o.,_ o.; o._; ,.o

_"xy Z

-- SI-F.ARSPC-GIk_ 1

--- 81.r=AR 8PECbI_ 2

.... SHEAR SPE_3

Figure 1!. Iosipescu shear test results: The plot shows the shear stress versus shear strain results for

the three iosipescu shear coupons cut from the web of the channel specimen.

Test Specimens 25
I00

75

50

25  ̧

/

//

/

./

./

0

o.oo........................... o.2s o.so o.................... 75 _.oo _.2s

_ 81=ECIMEN 1

--- TENSION _ 2

Figure 12. Tension test results: The plot shows the stress versus strain results for the two tensile

coupons cut from the cap flange of the channel specimen.

Test Specimens 26
Table2. Engineering constantsfromquasi-isotropic coupontests

Property Experimental Analytical

Extensional Modulus E_

In-plane shear modulus G,_

7.361 x 106 psi

2.655 x 106 psi

7.423 x 106 psi

2.862 x 106 psi

Error a

0.84 %

7.79 %

(Analysis - Exp)/Exp x 100

Test Specimens 27
Chapter 3

Experimental Apparatus and Tests

The experimental scope of this research includes both dynamic and static tests. The

dynamic portion of the experimental program employs several test methods to obtain

the natural frequencies and the associated mode shapes. The static phase of the research

program involves subjecting the specimens to combined loads at the midspan and

measuring the resulting displacements at the midspan. After the dynamic and static tests

were completed, the I-section specimen was instrumented with strain gages and prepared

for a quasi-static crush test. The experimental apparatus and test procedure for each

type of test will be discussed independently. The dynamic tests will be discussed first

followed by the static tests and the crush test.

Experimcntal Apparatus and Tests 28
Dynamic Tests

This section presents the experimental apparatus and procedures involved in the

dynamic tests. The first set of tests were run with free-free boundary conditions and used

an air shaker to excite the specimens. After these tests, the ends of the specimens were

secured in aluminum end fixtures using Hysol 934 potting compound. The remaining

tests were run with clamped-clamped boundary conditions. Each set of tests involved

exciting the specimens radially and laterally in order to excite both in-plane and out-ofplane modes. The tests are differentiated by the boundary conditions used. The free-free

frame tests will be discussed first followed by the clamped-clamped frame tests.

Free-Free Frame Tests

The free-free frame tests were conducted using the facilities at the Landing and Impact Dynamics Branch at NASA Langley Research Center. The experimental setup for

the free-free air shaker tests is shown in Fig. 13. The specimens were hung by elastic

bands at two points to emulate free-free boundary conditions. These tests used an air

shaker to excite the specimens with pulses of compressed air. The frequency and magnitude of the excitation were variable. The air shaker was a portable unit that required

no physical attachment to the specimens. Thus, the air shaker could be oriented in different directions to excite both in-plane and out-of-plane modes.

The response of the specimens was monitored with a hand held velocity probe. The

response signal from the probe was monitored visually on an oscilloscope. The probe

detected in-plane motion when oriented radially and out-of-plane motion when oriented

perpendicular to the plane of the frame. A natural frequency was isolated when the

Experimental Apparatus and Tests 29
(I

t:D

t--

;X

7-O

•--4 i'_

ooa

c_

G-)

:;o

>,

--o

.:E

_11..... iiil

Figure 13. Free-free flame vibrational test setup: The specimens were hung by elastic bands at two points and excited at one end by an air shaker.

The response of the frames was monitored using a velocity probe and an oscilloscope.
oscilloscope displayed a strong, clean, constant sinusoidal signal. A natural frequency

was tuned in nmch the same way that a radio station is tuned in. Once a natural frequency was isolated, a strobe light was used to to determine the frequency of excitation.

The specimen was excited at that frequency while the mode shape was studied and the

nodes located. The frequencies for the rigid body modes were much lower than the frequency of the first vibrational mode indicating that the boundary conditions were

achieved satisfactorily. The rigid body modes include in-plane and out-of-plane pendulum and rotational modes.

Clamped-Clamped Frame Tests

The clamped-clamped frame tests were conducted using the facilities in the Department of Aerospace and Ocean Engineering at Virginia Polytechnic Institute and State

University. The objectives of these tests were to determine the natural frequencies and

node locations of the clamped-clamped resonant modes and to obtain information on

the dynamic response of the frames over a range of frequencies in the form of frequency

response plots. A structural analyzer was used to generate the frequency response plots.

Based on the frequency response plots, a signal generator was used to excite the individual modes enabling the location of the nodal positions to be determined.

The clamped-clamped tests used a Synergistic Technology Incorporated (STI)

model 11/23 structural analyzer to generate the excitation signal and process the response data. The structural analyzer employs two interacting software packages set up

on a VAX minicomputer to control the excitation signal and to collect and process the

data. A flowchart of the test procedure is presented in Fig. 14. The excitation signal is

routed through an amplifier to an electromagnetic shaker. The electromagnetic shaker

Experimental Apparatus and Tests 31
Structural

Analyser

Actuator

Force

Output

Response

Signal

[

Force

Gage 2

I_e_uen_1

)

roxirnity

robes

Figure 14. Structural analyzer test flowchart: The structural analyzer collected force and response

data from the force gage and proximity probes. Fast Fourier Transforms of the force and

response data were calculated. Output displacement was divided by input force and plotted

on a log scale versus frequency in the form of frequency response plots.

Experimental Apparatus and Tests 32
is attached to the specimen via a sting and a piezoelectric force gage. The force gage

monitors the amplitude and frequency of the input force and sends this information to

the structural analyzer. The response of the frame is detected by Bently-Nevada noncontacting magnetic proximity probes which measure the displacement of small magnetic targets. A typical arrangement of the electromagnetic shaker, force gage, and a

proximity probe is presented in Fig. 15.

The operator prescribes a frequency range and step size for a given test. The structural analyzer sweeps through the frequency range, exciting the specimen at each frequency for fifty cycles and samples data fiom the force gage and proximity probes. The

structural analyzer calculates Fast Fourier Transforms (FFT's) of the collected force and

displacement data, divides displacement FFT by the force FFT and plots this ratio on

a log scale versus frequency in hertz.

]'he force actuator and proximity probes were oriented in different directions in order to excite and detect both in-plane and out-of-plane modes, respectively. ]'he radial

orientations for the force actuator and proximity probes are shown in Fig. 15. The outof-plane orientations for the shaker and proximity probes are shown in Figs. 16 and 17.

The specimens were clamped to a massive steel structure using C-clamps in an effort to

obtain clamped-clamped boundary conditions. The steel structure was shimmed and

leveled to provide a stable test platform. The shaker and the proximity probes were hung

from separate steel structures isolating them from the test specimen.

The structural analyzer results are presented in the form of frequency response plots

and phase angle diagrams. A plot was generated for each probe in each test. The plots

for an in-plane test of the I-section specimen are shown in Fig. 18. Spikes in the frequency response plots, accompanied by shifts in the phase diagram, indicate resonant

modes. The structural analyzer tests provided excellent values for the natural frequencies

Experimental Apparatus and Tests 33
O3

rr-"

-g

C. o'_

0

G_

Figure 15. Experimental apparatus for clamped-clamped dynamic tests: The electromagnetic shaker inputs a radial excitation force through the

force gage to the specimen. A proximity probe monitors the radial response of the test specimen.
tll

>

c_

:>

Z_

rq '

O

Figure 16. Experimental apparatus for clamped-clamped dynamic tests: l_e electromagnetic shaker inputs a lateral excitation force through the

t_ force gage to the specimen.
I20

r-

>

O

zo

! Z

i..rl f-_

0

_J

>

-0

Figure 17. Experimental apparatus for clamped-clamped dynamic tests: Two proximity probes are used to detect radial and lateral motion.
of the resonant modes, however, the test procedure made it impossible to study the mode

shapes of the modes associated with the spikes. It is impossible to identify the type of

mode or the nodal locations from the structural analyzer test data. A signal generator

was used to identify the modes and locate nodal positions.

The signal generator was used to excite tile specimens at the frequencies corresponding to the spikes in the frequency response plots. The signal generator sent the

signal to the electromagnetic shaker which excited the specimen. The signal from the

force gage was not monitored during these tests. The response signals from the proximity

probes were monitored visually one at a time on an oscilloscope. The mode was tuned

in by monitoring the response signals on the oscilloscope and adjusting the frequency

of excitation similar to the air shaker tests. Once a mode was isolated it could be excited

indefinitely enabling mode identification and the location of nodal positions.

Static Tests

This section discusses the test apparatus and procedures used in the static portion

of the experimental program. The objective of these tests was to determine the flexibility

matrices which relate radial, lateral, and twist displacements (W,V,ch) to the radial, lateral, and torsional loads (P,Q,T) at the midspan of the test specimens. The

displacement-load relation is

W O_ll

= 521

531

522 523 Q

532 533 T

{3.1)

Experimental Apparatus and Tests 37
C,O

! l I 1 I I I I I ! I I I I I I I] I i I 1 I I I 1 I I I I I I I i I I I

" ' A ' 0" OFF CMTR

60.62n r. _ / ", IM PLANE ."

06_U t i t t i if it I I I I I I I i t i I i l i i i t i t I : 'I I l I I I I I I • I

_,_. &,. .. __lOO. 0-_,, 2oo. 0 __,._ __.3oo.o, 400. N, v

ii z z z I z z z I i z t i z z z z I I I z I z'z 1' I'-i z I z z z z z i-z z" t-]

30.61 13" OFF CMTR -,

= IN PLANE =

--_ 12.86uI _ "7 , ' ' ' '

I t i i i , t I i I i t t t i 1 4 I i I i t I I I I l I I [ t i I i I t I i I I._

.,..,i

C_ _. ,,.Al,,,,4f,,,lw%, 0 1 " _ 100.0. _ "vw\ \.-7TA. L-, _%W_h,200.0 Jk '41..,f t_/_V_',,'_l'v ^I Y _.300.0_/ o ' ttJ'__'th V_7400.5,.d _ " 7 3,# #,4\

I I I I I I I I I I I I I I l I I I I I I I I I I I I I I .l I I l I I I I I

37.27n_. A ,,i A 5" OFF CMTR

'_.Ax / _.4_U)'_,/'I., ` .,J'l,_, ,_ OUT of" PLANE

0.0 100.0 200.0 300.0 400.5

Frequency (Hz)

Figure 18. Frequency resFinse plots for the I-section specimen: Frcquerlcy response plots for die I-section specimen under radial excitation. Spikes

indicate natural frequencies.
where c% are the flexibility influence coefficients.

Two sets of static tests were necessary to generate all of the flexibility influence coefficients. The in-plane tests loaded the specimen with a radial force and a torque, thus

facilitating calculation of the first and third columns of the flexibility matrix. The outof-plane tests loaded the frames with a lateral load and a torque, enabling the calculation

of the second and third colunms. In both sets of tests, the frames were clamped to a

massive steel structure which was shinuned and leveled to provide a consistent test

platform. The in-plane static tests will be discussed tirst followed by the out-of-plane

tests.

hi-Plane Static Tests

The in-plane static tests generated a radial load by hanging twenty pound calibration weights from the specimens using a load fixture attached to the specimen. The

load fixture and the coordinate system are shown schematically in Fig. 19. The weights

were hung from a steel cable which was positioned between two nuts on a threaded rod.

By moving the nuts along the threaded rod, the load could be traversed across the cross

section, thus changing the torque applied to the cross section. The torque for each test

is the product of the radial load (P) and the moment arm (D). The moment arm is the

distance along the threaded rod from the load application point to some point of reference where the torque is taken to be zero. It is convenient to define this point of reference as the point where a radial load causes no twist and no lateral deformation. This

point is analogous to the shear center of a straight beam.

Photographs of the load fixture and the in-plane static test apparatus are shown in

Figs. 20 and 21, respectively. Displacement measurements were taken at the midspan

Experimental Apparatus and Tests 39
,'IT

ill

= V

 JJJ JJJJJJJJ 

P

Z

T:P.D

_Y

Figure 19. Load fixture and Coordinate system: The load fixture applied radial, and torsional loads

to the specimen. The applied torque was the product of the radial load (P) and the moment

arm (D). Radial _V), lateral ('V), and twist (&) deformations were measured with respect

to the centroidal coordinate system shown.

Experimental Apparatus and Tests 40
using dial indicators which have a range of one inch and a resolution of .0005 inches.

Two dial indicators were oriented radially on top of the load fixture and a third was

oriented laterally on the side of the load fixture. Assuming that the cross section of the

test specimen and the load fixture move as a rigid body, three measurements are sufficient to define the deformed position of a body moving in a plane. The data from the

dial indicators were resolved into lateral and radial displacements of the centroid and the

twist of the cross section. The data reduction scheme used to obtain the centroidal displacements and a sample of the reduced data are presented in Appendix B. The dial indicators were attached to separate steel frames isolating them from the specimens.

Readings were taken each time a weight was loaded or unloaded and the values were

averaged. The frames were loaded radially to 120 pounds.

For the in-plane static tests, the displacement-load relationship reduces to the following equation because the lateral load (Q) is assumed zero. Thus, the middle column,

cq2, e22, and Cqa, cannot be calculated from the in-plane tests.

W r °_11 {x13

{v}:[ + _31 0_33

{;} (3.2)

The nonlinearities of this problem are most prevalent when the frames bend out of

plan'e. In order to minimize the nonlinear response, the maximum loads were selected

such that the maximum lateral displacement was limited to one half inch. This limitation

effectively dictated the maximum allowable torque since the maximum radial load was

already set. The torque is the product of the radial load and moment arm, thus the moment arm was adjusted to keep the lateral displacement within the allowed limit.

Experimental Apparatus and Tests 41
0

.... .2.

r" '

J_ ;'_

Figure 20. Load fixture: Dial indicators were used to measure the displacements of the midspan. A data reduction scheme was used to obtain

the centroidal displacements.
.!

oi

ttl

r

c)

O

c-)

,_ Figure 21. In-plane test apparatus: Twenty pound calibration weights were hung from ttle load fixture to generate the radial load. Tile torque

was changed by moving the line of action of the radial load relative to the cross section.
The point of zero twist marked the center of the test range for each specimen. Since

tile l-section specimen was stiffer then the channel specimen with respect to torsion, the

maximum moment arm for the I-specimen was correspondingly larger resulting in a

larger test range. Many tests were run with the load application point traversing through

the test range subjecting the tests specimens to positive and negative torques. Tile results of the static tests were plotted as load versus displacement for various values of the

moment arm. The flexibility influence coefficients were calculated from this data.

Out of Plane Static Tests

The out-of-plane tests required slight changes in the test apparatus and procedures.

Tile out-of-plane tests involved loading the specimens with a lateral load and a torque.

The load fixture was modified (Fig. 22) so that loads could be applied perpendicular to

the plane of the frame. The load fixture had a tendency to pull away from the test

specimen requiring the load fixture to be tied to the specimen, thus, forcing the load

fixture and the test specimen to deform as a rigid body. The frames are very compliant

laterally requiring smaller loads to reach the limiting lateral displacement. The moment

arm reference points for the out-of-plane tests were chosen as the centroid of the cross

sections.

Photographs of the modified load fixture and tile test apparatus for the out-ofplane static tests are shown in Figs. 23 and 24, respectively. Calibration weights were

hung from a piece of twine which was routed over a pulley and attached to the load

fixture. The load application point was changed by traversing two nuts along a threaded

rod similar to the in-plane tests. Changing the load application point effectively changed

Experimental Apparatus and Tests 44
!

I

I

I

I

i

I

Z

W

]

L

yY

D

V--

=O'D

0

r

Figure 22. Schematic of modified load fixture: Load fixture was modified for the out-or-plane tests to accept lateral loads. The torque (T) was

taken to be the product o[" file lateral load (Q) and the moment arm (D) where the moment arm was file perpendicular distance from

the ceatroid to the line of action of the lateral load.
the relative torque applied to the specimen. 'file pulley was mounted on a traverse so

that the twine could be kept level, thus, keeping the load perpendicular to the frame.

For tile out-of-plane static tests, the displacement-load relationship reduces to the

following equation because the radial load (P) is assumed zero.

W

{vt

4,

_12 _13

0{22 0{23

0{32 0{33

(3.3)

Thus, the first column, 0{., 0{a_,and 0{31, cannot be calculated from the out-of-plane tests.

The torque (T) is tile product of the lateral load (Q) and the moment arm (D). Three

dial indicators were used to measure the displacements of the specimen at the midspan.

The dial indicators were arranged in the same manner as in the in-plane tests, thus, the

same data reduction scheme was used.

Maximum lateral loads of 4.5 and 2.3 pounds were used for the I-section and

channel section specimens, respectively. Readings were taken from the dial indicators

each time a weight was loaded or unloaded. The resulting data was then averaged. After

each test the moment arm was incremented by one-eighteenth of an inch (one turn on

threaded rod) and the procedure was repeated. The test range for the out-of-plane tests

was larger than the test range in the in-plane tests, but the loads were much smaller

yielding smaller torques.

Experimental Apparatus and Tests 46
.--.I

1

Figure 23. l'hotograph of modified load fixture and dial indicators: Dial indicators were used to measure the displacement of the load fixture at

the midspan. The load fixture was tied to the specimen to force them to deform as a rigid body.

t_

I--

>

O

>

I

o

::D

"O

.:IZ

o

r|
/" f i"

Figure 24. Out of plane test apparatus: Load path for the out-of-plane tests. The load was applied perpendicular to the frame by hanging weights

from a line looped over a pulley.

l--

C)

:E 0

t:3:0

"--4 rvt

C)

¢D

':E
Crush Test

The crush test is part of the ensuing phase of this research and only has a tangential

application to this study? ]he objective of the crush test was to obtain experimental

data on the large deflection, failure, and post failure responses of the frame. To obtain

this data, the I-section specimen was loaded radially through successive failure events.

The current research is only concerned with the linear small deflection response of the

frame, however, since the crush test loading is similar to the in-plane static tests, the

crush test provided a second opportunity to determine the flexibility coefficient cq>

A photograph of the test apparatus is shown in Fig. 25. ]'he frame was restrained

from deforming laterally or twisting at the midspan and the potted ends were bolted to

an I-beam to simulate clamped-clamped boundary conditions. The crush test was

slightly different from the in-plane static tests where the midspan was unrestrained and

the potted ends were clamped to the l-beam. The frame was instrumented with strain

gages to measure the strain distribution. A string potentiometer was used to measure the

radial displacement. A Tinius-Olsen test machine was used to apply the load. For the

midspan restrained specimen, the displacement-load relation is

W F 511 Ctl2 513

{0}=[....

0 531 532 533

P

T

(3.4)

where V and q5 have been restrained and Q and T are not necessarily zero. Thus, if

_,2 and _n are sufficiently small the radial displacement is simply the product of the raThe static crush test was conducted at Virginia Tech in cooperation with Mr. E. Moas, graduate research

assistant, and Professor O. H. Grillin. This project is also supported by the Landing and Impact Dynamics Branch, NASA Langley Research Center.

Experimental Apparatus and Tests 49
ORIGINAL _,_.me

BLACK AND WHITE Pi-;Oi-OGRAPH

G,

E

O

m

©

o_I--

i'm

>-.

E

.-9

cz_

o

tm

E

t.-

©

r..

[-

¢.lj

...i

i...

,,4

0__._

51)
dial load and _11' The flexibility coefficient cq_ was obtained directly from the loaddisplacement plot.

Experimental Apparatus and Tests 51
Chapter 4

Experimental Results

Data from the dynamic testing is presented first followed by the static test data. The

dynamic test data includes the natural frequencies and node locations of the specimens

for free-fi_ee and clamped-clamped boundary conditions. The static test data is presented

in plots of load versus displacement. The flexibility influence coefficients were calculated

from this data.

Dynamic Test Data

In general, the vibrational response of the frames can be divided into in-plane and

out-of-plane motion. The vibrational response of the I-specimen completely decouples

into in-plane and out-of-plane modes. All the modes of tile channel specimen exhibit

coupling between in-plane and out-of-plane motions, but the modes were clearly dominated by one type of motion. The photograph in Fig. 26 is representative of the coupled

Experimental Results 52
response observed in the modes of the channel section specimen. "[he photograph shows

the first free-free in-plane flexure mode of the channel specimen. ]'hough the cross section experiences some lateral and twisting motion, the dominant motion is radial. The

photograph in Fig. 27 shows a pure out-of-plane motion indicative of the uncoupled

response displayed by the I-section specimen. The photographs in Figs. 26 and 27 were

taken during the free-free vibrational tests using a strobe light which was tuned to twice

the excitation frequency. Using this technique the cross section was illunfinated at either

extreme of the motion, thus, capturing the characteristic motion of that mode.

A mode was identified by two characteristics, the dominant motion and the number

of nodes in the mode shape. The nodes of an in-plane mode are defined as those points

where the radial motion is nearly zero. This does not mean that the out-of-plane displacement is zero there, though in general it is small too. Likewise, the nodes of an

out-of-plane mode are defined as those points where the out-of-plane bending and

twisting motions are nearly zero.

Three distinct types of modes were observed in the free-free vibrational tests,

whereas only two types of modes were observed in the clamped-clamped tests. For the

free-free tests, motions associated with in-plane bending, out-of-plane bending, and

torsion were observed at different frequencies. For the clamped-clamped tests, motions

associated with in-plane bending and coupled out-of-plane bending and torsion were

observed at different frequencies. Clamped end conditions caused a stronger coupling

between out-of-plane bending and torsion than is present in the free end case. For either

end condition, the initial curvature of the specimens statically couples the out-of-plane

bending moment and the torque. However, the magnitudes of these actions are likely to

be much less for the free-free case than for the clamped-clamped case. This may be the

reason for the relatively uncoupled motions associated with out-of-plane bending and

Experimental Results 53
.... ";" : ,'_S_%8

Figure 26. First free-free in-plane mode of the channel specimen: The motion shown in the photograph is typical of the coupled response demonstrated by the channel section specimen.
t.lt

t._

tXi

[-.-.

=o

:EZ

F"

C_

,_[ [_'I

O

if)

;:o

"n

,:E

Figure 27. Second flee-flee out-of-plane mode of the i-specimen: The cross-sectional motion shown in the photograph is purely lateral. Thus, the

l-section decouples in-plane and out-of-plane response.
torsion observed in the free-free tests. For example, out-of-plane bending dominates tile

motion in the mode at 26.3 Itz for the free-free I-specimen shown in Fig. 27. The same

I-specimen vibrating at 57.2 Itz, as shown in the photograph in Fig. 28, exhibits a

torsionally dominant motion.

Natural frequencies and node locations for the free-free in-plane, out-of-plane, and

torsional modes for both specimens are presented in Table 3. The natural fi'equencies

are in hertz and the node locations are measured in radians from the midspan. The node

locations are synm_etrically located about the midspan. The plot in Fig. 29 compares tile

in-plane, out-of-plane, and torsional natural frequencies of the two specimens. The

number of nodes is plotted on the horizontal axis and tile natural frequency is in hertz

on tile vertical axis. The fiequencies associated with the in-plane modes of the Ispecimen are higher than the frequencies associated with the in-plane modes of the

channel specimen. This indicates that the I-specimen is stiffer radially. The data ['or the

two specimens show increasing frequencies with increasing number of nodes in the

modes. Tile in-plane natural frequencies for the channel specimen are consistently 15-25

% lower than the corresponding l-specimen fi'equencies. The curves in Vig. 29 (B) correspond to the natural frequencies of the out-of-plane flexure and torsional modes of

both test specimens. The torsional natural frequencies associated with the I-specimen

are consistently higher than the corresponding channel modes indicating that the Ispecimen is stifler with respect to torsion. The natural frequencies for the out-of-plane

flexural modes for the two specimens coincide.

The results of the clamped-clamped vibrational tests for the I-section and channel

section specimens are presented in Table 4. The frequencies are in hertz and the node

locations are in radians measured from the midspan. The clamped ends are counted as

two nodes. Thus, two nodes is the minimum number in a mode for the clamped-clamped

Experimental Results 56
F-

>

>

Z 0

U_

!Z

--t r""

rn

0

fD

"c,

Figure 28. First free-free torsional mode of the I-specimen: The motion of the cross section is predominantly torsional with a small amount of

lateral motion compared to the out-o['-plane flexure mode ill I:ig. 27. The ['tee-free boundary conditions reduce tile e[tbct of the geometric coupling between torsion and out-of-plane bending.
Table 3. Natural frequencies and node locations of the free-free modes

I-specimen

In-plane

Out-of-plane

Torsion

Channel specimen

In-plane

Out-of-plane

Torsion

No. of Freq

Nodes Hz

2 25.8

3 78.3

4 151.0

5 259.0

6 388.0

3 8.1

4 26.3

3 57.2

4 97.5

5 146.0

6 200.0

2 18.4

3 65.0

4 128.0

5 205.0

6 280.0

3 7.4

4 26.1

3 47.3

4 87.5

5 137.0

6 188.0

-1.23

- 1.23

Node Loealions

Radians from Midspan

- 1.22

-1.30

-0.61 0.63

- 1.09 -0.02 1.05

-0.43 0.43

-0.70 -0.01 0.66

1.23

1.29

-1.30

-1.16 0.02 1.16

-0.48 0.48 1.28

-0.75 0.02 0.72

-0.98 -0.30 0.30 0.95

-1.08 -0.51 0.01 0.53 1.08

-0.92 -0.29 0.29 0.87

-1.22

-1.31

-0.65 0.59

-1.I0 -0.00 1.11

-0.49 0.51

-0.75 0.01 0.75

1.23

1.32

- 1.28

-1.14 -0.01 1.15

-0.45 0.46 1.30

-1.07 0.02 1.01

-1.19 -0.47 0.44 1.16

-1.20 -0.68 0.01 0.66 1.18

-0.92 -0.29 0.29 0.87

1.20

1.20

Experimental Results 58
_jqO0 "t

300,

/

0

o

NO, OF NODE8

---- CHANNR.

(A)_-P_NE NODES

25O

2oo p

iY

| 50 J/i •

o 2 i ........ i ........

lqO. 01='NOOE8

•.u _ I

•--- OUT-OF-PLANE G

N. OUT-OF-PLAI_ i

(2) TORSIONAL AND 0UT-0F-PI.ANE MODES

Figure 29. Comparison of free-free natural frequencies: The in-plane and out-of-plane natural frequencies for both test specimens arc shown in the two plots. The number of nodes in a

mode is plotted on the horizontal axis and natural frequency in hertz on the vertical axis.

Experimental Results 59
]'able 4. Natural frequencies and node locations of the clamped-clamped modes

I-specimen

In-plane

No. of Frcq

Nodcs Hz

3 57.8

4 127.2

5 233.0

6 350.6

7 490.2

8 617.3

Node Locations

Radians from Midspan

-1.57 0.00 1.57

-1.57 -0.57 0.58 1.57

-1.57 -0.72 0.00 0.75 1.57

-1.57 -0.94 -0.26 0.28 0.94 1.57

-1.57 -1.01 -0.49 -0.01 0.48 1.01 1.57

Out-of-plane 2

3

4

5

6

7

8

9

10

8.2 -1.57 1.57

29.8 -1.57 -0.04 1.57

64.3 -1.57 -0.34 a 0.49 1.57

110.4 -1.57 -0.62 -0.01 0.58 1.57

157.8 -1.57 -0.85 -0.28 0.25 0.88 1.57

164.7 -1.57 -1.02 -0.42 a 0.08 0.54 1.08 1.57

207.4 -1.57 -1.23 -0.74 -0.24 0.22 0.73 1.20 1.57

254.7 -1.57 -1.28 -0.80 -0.37 0.07 0.43 0.85 1.31 1.57

314.7

Channel specimen

In-plane 3

4

5

6

7

44.6

99.1

176.0

276.7

413.0

-1.57 -0.01 1.57

-1.57 -0.58 0.61 1.57

-1.57 -0.84 -0.14 0.82 1.57

-1.57 -0.98 -0.29 0.28 0.94 1.57

Out-of-plane 2

3

4

5

6

7

8

9

10

7.4 -1.57 1.57

25.4 -1.57 0.09 1.57

55.4 -1.57 -0.40 0.35 1.57

91.1 -1.57 -0.60 -0.05 0.60 1.57

148.8 -1.57 -0.85 -0.28 0.27 0.83 1.57

155.2 -1.57 -1.00 -0.29 -0.01 0.35 0.90 1.57

187.7 -1.57 -1.26 -0.76 -0.26 0.25 0.76 1.26 1.57

216.0 -1.57 -1.26 -0.84 -0.37 0.06 0.37 0.72 1.20 1.57

265.0

Node near the force actuator

Experimental Results 60
specimen tests. In general, the nodes are distributed symmetrically about the midspan

except when a node falls close to the force actuator, which tends to distort the mode

shape.

The in-plane and out-of-plane natural frequencies for both specimens are compared

in Fig. 30. The response of the two specimens is qualitatively the same. The in-plane

natural frequencies of the I-specimen are consistently higher. The first out-of-plane

mode for the two specimens is nearly identical, which is similar to the coincident outof-plane flexural modes observed for the two specimens in the free-free tests. The natural

frequencies of the I-specimen for subsequent out-of-plane modes are higher than the

corresponding channel specimen frequencies. The reduction of the increase in frequency

with increasing number of nodes in Fig. 30 (B) at six nodes is explained by a softening

of the end restraint conditions.

Static Test Data

The experimental data from the static tests is presented in plots of load versus displacement. The flexibility influence coefficients were calculated using a least squares

routine which fitted the best plane to the data. The load-displacement plots will be discussed first followed by the data reduction scheme and the flexibility influence coefficients.

Experimental Results 61
lq0o

O

o

0

o ......... i

/

/

J

NO. OF NO05

144B_

(A) IN-PLANE MODES

D

/

/s

......... tO

cI

I

I

YjI

NO. _ NaZB

IIQ4

(]3)OUT-O_-P_ mODES

Figure 30. Comparison of clamped-clamped natural frequencies: The in-plane and out-of-plane natural frequencies for both specimens are compared in the two plots. The number of nodes

in a mode is plotted on the horizontal axis and natural frequency in hertz on the vertical

axis.

Experimental Results 62
Load-Displacement Plots

In the ensuing presentation of the load-displacement plots, each figure will present

the data for both specimens. The data associated with the I-specimen will be presented

in the (A) part of each figure and tile data for tile channel specimen will be presented in

the (B) part. The data from the in-plane static tests will be plotted as radial load versus

radial, lateral, and twist displacements for various values of the moment arm. The outof-plane static test data will be presented as lateral load versus radial, lateral, and twist

displacements for various values of the moment arm. The applied torque in either test

is the product of the applied force and the moment arm. The in-plane test data will be

presented first followed by the out-of-plane test data.

The plots of radial load versus radial displacement are presented in Fig. 31. The Ispecimen data is linear through the maximum radial load and shows no sensitivity to the

applied torque. This indicates that the flexibility coefficient cq3 is small. "File channel

specimen data is linear through a radial load of sixty pounds. The spread in the curves

for the channel specimen indicates that the channel specimen's flexibility coelticient _3

is not zero.

The plots of radial load versus lateral displacement are shown in Fig. 32. The Ispecimen data is symmetrically arranged about the zero torque line (D = 0). The fan

shape indicates that the flexibility coefficient _23 is nonzero. Nonlinearities appear in

some of the curves, but the data corresponding to smaller moment arms is linear through

a radial load of sixty pounds. The channel specimen data is not symmetrically arranged

about the zero torque line. This indicates that the channel specimen's value for flexibility

coefficient cc23is dependent on the sign of the torque. The curvature in tile data for the

channel specimen is more pronounced than in tile I-specimen data.

Experimental Results 63
150'

120,

o........ ._ ....... .'_........ .i_ ....... .'2

----- D = .2083"

_--_ D = .0417"

--.-. D -- O41"7"

----. D -- .2083"

ON)

H-. D • .12S0"

•_.D= .0000"

•I-. D -- .1250'

(A) I-SPECIM_

, - ....... , ......... , ...... . ...........

-----D= .0417" ----=D= .0206"

--.D= .00130" -----D=-.O20ff

----- D -- .0417" ----- D -- .0e25"

(B) _EL SPEC_EN

Figure 31. Radial load versus radial displacement: Radial displacement in inches is plotted on the

horizontal axis and radial load in pounds on the vertical. The l-section data is linear and

insensitive to torque. The channel specimen exhibits nonlinear response and a sensitivity

to torque.

Experimental Results 64
150.

1209O

0

o. 75 -.5 -._ 0 .25 .5 .75

LA'rB:IAL I]ER.qcB_qT (V)

----.,.D • .2083" _ D - .1250"

-,-,-.,.D- ,0417" ---,--D= ,0000"

D -- .0417" _ O -" .1250"

,_,-_D ,,- .20_3"

(A)]-SPECn4EN

t20

9O

SO

O.

-. 75 -.5 -.25 0 .25 ,5 .75

LAllSCq. DIB:q.N_B_qT (Y)

•---- D- J)417" ----.D=.020¢

,-,--O- J_00" --,.-.D.,-.0208"

•-.-- O -- .0417" --,,-.O -- ,0825"

(B) CHANNELSPECUMF__

Figure 32. Radial load versus lateral displacement: Lateral displacement in inches is plotted on the

horizontal axis and radial load in pounds on the vertical. The 1-section data is symmetrically arranged about the zero torque line. The channel data is not symmetrically arranged

indicating ,,23 is dependent on the sign of the torque.

Experimental Results 65
Figure 33.

i

-0,075 -0. 050 -0.0_3 0. 000 0. 025 0. 050

nvermr^m. .,,mine

.-,--D= .2083" --- D= .1250"

-,-,-.D- .0417" _.,-D= .0000"

•.-*-* D ,,," .0417" ----,, D ,,'- .1260"

•.-.,-..D =- .20_"

(A) I-SPECIMEN

0.075

150.

120gO

30

0

-0,075 -0.050 -0.025 0.000 0.025 0.050 0.075

TWIST ROTATIICN (_)

----. D= ,0417" ---.D-_

..--.D- ,0C00" ,.-,.--D=-.0208"

----- D .,- ,0417" --- D .,- .0625"

(B) CHANINk-_LSPECIMF___

Radial load versus twist displacement: Twist displacement in radians is plotted on the

horizontal axis and radial load in pounds on the vertical. The l-section data is symmetrically arranged about the zero torque line. The channel data is not symmetrically arranged

indicating _33 i_ dependent on the sign of the torque.

Experimental Results 66
The plots of radial load versus twist are presented in Fig. 33. The I-specimen data

is symmetrically arranged about tile zero torque line indicating a consistent value for

flexibility coefficient 0{33. Generally, the data is linear through the highest radial load.

The data for the channel specimen is not symmetrically arranged indicating that the

channel's torsional stiffness 0{33is dependent on the sign of the torque. The curves are

linear through a radial load of sixty pounds.

The plots of lateral load versus radial displacement from the out-of-plane static tests

are presented in Fig. 34. The I-specimen data shows significant radial displacements indicating that tile flexibility coefficient 0{12is nonzero. This is inconsistent with the

vibrational experimental data since a nonzero value of 0{12couples the in-plane and outof-plane responses. This inconsistency of a nonzero 0{_2is thought to be due to a geometric nonlinear effect of the reference arc of the frame. For inextensional response,

an out-of-plane displacement in either direction at midspan would cause a radially inward displacement. This is analogous to a cantilevered beam subject to a transverse load

in which the axial displacement at the tip is directed inward no matter the sense of the

applied load. The radial displacement for the channel specimen is also nonzero. It is

difficult to say how much of the radial deformation is a function of the geometric nonlinearity and how much is due to the coupling caused by the asymmetric cross section.

Plots of lateral load versus lateral and twist displacements are presented in Figs. 35

and 36, respectively. The I-specimen data for the two figures is linear through the maximum lateral load. The data for the channel specimen exhibits very slight curvature.

The in-plane and out-of-plane static tests were necessary to generate all three columns of the flexibility matrix. The frames are stiffer in the radial direction enabling

higher loads, hence larger torques. The smaller torques in the out-of-plane tests resulted

Experimental Results 67
[

0

0 ........ _oos " .o':0....... _o'lS....... .o2

FU_I_ _61_.AC81_I" (Wl

--..--D - -.e,.85" ,--4 D - -.518"

-,.-.--D- -.352" ----.- D - -.185"

----- D - -.018" ----- D ,, .148"

(a),-sPE_

Figure 34.

0 .tX)S .OI .OrS

RN_ IZSR.N_4m'r (W)

•----D, -_186" ,-,4 D ,, -_'18"

•-.-.D--_:2" ----- D- -.186"

•-.-- D - -.018" ----- O - .148"

(B) CIIANNI_ILSPECIMEN

.O2

Lateral load versus radial displacement: Radial displacement in inches is plotted on the

horizontal axis and lateral load in pounds on the vertical. The I-section data is nonlinear

and nonzero which is inconsistent with previous observations. Channel data is linear and

shows substantial radial displacements.

Experimental Results 68
5,

.a3

Figure 35.

-----D - -,i_5" ,-,,-,,D ,, --,618"

•--,-,D--,352" _ D - ",'185"

•-,-- D - -.018" -.---.-D ,, ,14_

(A) I- SP_.CI_EN

(B) CII_NEL SPECIM_

Lateral load versus lateral displacement: Lateral displacement in inches is plotted on the

horizontal axis and lateral load in pounds on the vertical. The l-section data is linear

through the maximum lateral load. The channel data exhibits some curvature. The data

points are closely spaced.

Experimental Results 69
q

0

Figure 36.

0._)

......... , ,- ....... . ........ , ........

O.Ot 0,02 0.03 0.0_1 0,05

!

TWIST ROTATION (_))

•----O - -J_86" ----,D - -.618"

D - -.352" ---t D - -.186"

---.-- D - -.018" ----- D - .148"

(A) I-SPECIMEN

....... ,. , ........ , ........ . ........

0.00 O.Ot 0.02 0.03 O. Oli O.OS

• -.618"

•-.-.D--.362" _ D- -186"

•---- D - -.018" ----- D - .148"

_B) CltANN_ SPECIMF_

Lateral load versus twist displacement: Twist displacement in radians is plotted on the

horizontal axis and lateral load in pounds on the vertical. The l-section data is linear

through a lateral load of three pounds. The channel data is slightly nonlinear.

Experimental Results 70
in smaller effects which were difficult to measure reliably. Thus, the data from the inplane tests is regarded as more reliable than the out-of-plane test data.

Data Reduction

For the in-plane tests, the load-displacement relations reduce to

{vt: 'v t ¢_ 531 0C3 3

(4.1)

Many tests for different values of the radial load and moment arm yield an algebraically

overdetermined system. This situation occurs frequently in experimental research when

more data are generated than would be required if absolute precision was possible. If

such precision was possible, only two tests would be required to determine the flexibility

coefficients.

The first of Eqs. (4.1) can be interpreted as a plane in the W, P, T space. Sinfilar

interpretations can be given to the second and third equations. The flexibility coefficients were calculated by fitting the best planes to the experimental data using a least

squares routine. The experimental data were treated in groups in order to better understand the response of the frames under different loads. The experimental data was

divided according to the value of the radial load and the algebraic sign of the torque.

Thus, flexibility coefficients were calculated for positive and negative torques for each

value of the radial load.

For the out-of-plane tests, the load-displacement relations reduce to

Experimental Results 71
W

Ivt

_12 _13

_22 _23

_32 _33

(4.2)

Similar to the in-plane tests, repetitive tests yielded an algebraically overdetermined

system. The relations in The first of Eqs. (4.2) can be interpreted as a plane in the W,

Q, T space. Similar interpretations can be given to the second and third equations. The

flexibility coefficients were calculated by fitting the best plane to tile experimental data

using a least squares routine. Flexibility coefficients were calculated for each value of

the lateral load.

In general, the in-plane tests measured the flexibility coefficients more consistently

than the out-of-plane tests, and the coefficients on the diagonal were measured more

consistently then the off-diagonal terms. With the exception of c_23and cq2 , tile offdiagonal terms were small in magnitude and had a significant amount of scatter associated with them. The _2a terms are significant because of the coupling between

out-of-plane bending and torsion due to the curved geometry.

No attempt was made to force the flexibility matrix to be symmetric. Enforcing

symmetry in the flexibility matrix would substantially alter the values of the off-diagonal

coefficients. The curved frames loaded at the midspan present geometric nonlinearities

and the test data indicates this. Forcing the matrix to be syrmnetric would misrepresent

the data and therefore tile response of the frames.

Flexibility Influence Coefficients

Experimental Results 72
Table 5. Flexibility influence coefficients fi_r I-specimen from in-plane tests

3

e"

m

Radial Load T' ,_. _,_ _z_ a,3 a3, az,

x 10 .3 x 10 .2 x 10-_ x 10 .4 x I0-* x 10 4

lbs +/- in/Ib 1/in-lb 1/ib 1/lb 1/ib in/lb

P = 20 0.875 0.136 0.092 1.750 0.415 0.086

P = 20 + 0.874 0.134 0.108 1.410 0.301 0.104

P = 40 0.843 0.134 0.101 0.930 0.265 0.741

P = 40 + 0.842 0.128 0.102 1.200 0.257 0.575

P = 60 - 0.831 0.137 0.105 0.415 0.228 0.422

P = 60 + 0.838 0.130 0.106 0.135 0.218 0.478

P = 80 - 0.823 0.136 0.110 0.200 0.175 0.178

P = 80 + 0.823 0.133 0.110 0.372 0.159 0.187

P = 100 - 0.814 0.138 0.116 -.056 0.081 -.188

P = 100 + 0.815 0. 141 0.121 0.969 0.096 -.284

Algebraic sign of applied torque

--..I
The flexibility influence coefficients for the I-specimen from the in-plane tests are

presented in Table 5. Flexibility coefficients are presented for radial loads from 20 to 100

pounds for both positive and negative torques. The second column indicates the algebraic sign of the applied torque. In general, the flexibility coefficients for the I-specimen

were unaffected by the sign of the torque.

The values for the flexibility coefficient _H get gradually smaller with increasing

value of the radial load indicating a stiffening in the response of the frame. The values

for flexibility coefficient cq3 obtained for negative torques are slightly larger than the

values obtained for positive torques, however, the difference in magnitudes is small. The

values for flexibility coefficient _23 become gradually larger with increasing radial load

indicating a slight softening in the measured response. The values for flexibility coefficients cq3, cq_, and cx2,are small and are inconsistently measured. Significant scatter in the

data associated with these coefficients make a reliable determination of their magnitudes

difficult. This indicates that their affect on the response of the frame is small. This is

consistent with the uncoupled vibrational response observed in the dynamic tests since

these coefficients couple the in-plane and out-of-plane static responses.

The flexibility influence coefficients for the channel specimen from the in-plane tests

are presented in Table 6. Flexibility coefficients are presented for radial loads from 20

to 100 pounds for both positive and negative torques. The values for flexibility coefficient cql are consistent with respect to the algebraic sign of the applied torque and with

respect to the magnitude of the radial load. The values for Cql for the channel specimen

are roughly twice the magnitude of corresponding values for the I-specimen. The channel specimen does not exhibit the stiffening response observed in the I-specimen. The

values for flexibility coefficients cq3 and c_23become steadily larger with increasing radial

load indicating a softening in the torsional resistance. Further, the values for cq3 and c_23

obtained for negative torques are consistently smaller than values obtained for positive

Experimental Results 74
Table 6. Flexibility influence coefficients fiJr channel specimen from in-plane tests

3

Radial Load T" a,, 0%3 %1 or,3 a_l a2,

x 10 -3 x 10 2 x 10'l x 10 -2 x 10 -4 x IO 3

lbs +/- in/lb 1/in-lb l/lb l/lb l/lb in/lb

P = 20 1.64 0.250 0.196 0.313 -0.332 0.049

P = 20 + 1.66 0.290 0.219 0.301 -0.232 0.164

P = 40 1.66 0.258 0.224 0.361 -0.248 0.079

P = 40 + 1.66 0.295 0.264 0.377 -0.154 O. 157

P = 60 1.65 0.284 0.271 0.391 -0.463 0.065

P = 60 + 1.66 0.285 0.308 0.411 -0.200 0.150

P = 80 i.65 0.323 0.341 0.442 -0.209 0.(/28

P = 80 + 1.66 0.332 0.394 0.492 -0.067 0.139

P = 100 1.65 0.394 0.457 0.526 -0.219 0.025

P = 100 + 1.66 0.454 0.548 0.655 -0.071 0.132

Algebraic sign of applied torque

tJl
torques. This indicates that the torsional stiffness of the channel specimen is a function

of the algebraic sign of'the torque. The values for flexibility coefficients _,3 and _3, differ

by two orders of magnitude and have opposite signs. The magnitudes of the values for

_-3, depend on the sign of the torque. The scatter associated with these values makes

reliable determination difficult. The values for flexibility coefficient a21 are a function of

the sign of the applied torque and tend to get smaller with increasing value of the radial

load.

The llexibility coefficients for the I-specimen from the out-of-plane tests are presented in ]'able 7. Coefficients are presented for four values of the lateral load. ]'he values for flexibility coefficient _22 are measured consistently for each value of the lateral

load. The values for flexibility coefficient _33 are also measured consistently, however, the

values are about 20 %% smaller than corresponding values obtained from the in-plane

tests. The values for flexibility coefficients cz2aand _32 are roughly the same magnitude,

however, their magnitudes are consistently 15-20 % smaller than the values for c% obtained from the in-plane tests. The values for flexibility coefficient _,3 are inconsistently

measured and the scatter associated with this coefficient is significant. The values for

flexibility coefficient _2 become steadily larger with increasing lateral load and are about

100 times larger than the values for _, obtained from the in-plane tests. As was discussed

previously, this is thought to be due to a geometrically nonlinear response similar to the

end shortening of a cantilevered straight beam.

]he flexibility influence coefficients for the channel specimen from the out-of-plane

tests are presented in Table 8. Flexibility coefficients are presented for three values of the

lateral load. ]'he values for flexibility coefficient _22 are measured fairly consistently. The

magnitude of the values are roughly twice the magnitude of the corresponding values for

the l-specimen indicating that the channel specimen is twice as compliant laterally. The

values for flexibility coefficient _3 become smaller with increasing value of the lateral

Experimental Results 76
Table 7. Flexibility influence cocfficicnts for I-specimcn from out-4_f-planc tests

§.

P..

e.

Lateral Load _22 _ o_z_ o%z ot,_ o_,2

x 10 2 x 10 -2 x 10 .2 x 10 _ x 10 2

lbs in/lb 1/in-lb 1/Ib 1/lb 1/lb in/lb

Q = 2.187 0.112 0.100 0.802 0.864 0.232 0.118

Q = 3.250 o. 113 0.111 0.874 0.877 0.368 o. 148

Q = 4.125 o. 112 o. 114 0.860 0.864 0.400 o. 173

Q = 4.500 o. 110 o. 105 0.595 0.857 0.324 0.181

',,,,4

"..4
load. This trend is opposite to the trend observed in the values for tile _3a from the inplane tests. The values for c_33from the out-of-plane tests are much smaller than the

values obtained from the in-plane tests. The values for flexibility coefficient c_23are relatively consistent and compare more favorably with corresponding values from the inplane tests. The values for c_3_are about 60 % of the values for c_23.The values for

flexibility coefficient cq2 are measured consistently and are much larger than the values

for _21 from the in-plane tests. It is believed that the larger values for al_ is indicative of

a geometrically nonlinear response.

The values for C_2aand c_33which were obtained from the out-of-plane tests are generally smaller than the corresponding values obtained from the in-plane tests. The fact

that the flexibility coefficients calculated from the two tests are not consistent is disturbing. The out-of-plane tests used very' small lateral loads and smaller torques. The

resulting displacements due to the torques were correspondingly smaller making them

more difficult to measure experimentally. Thus, the values for _3 and c% obtained from

the in-plane static tests are considered more reliable.

Crush Test

The load displacement curve from the quasi-static crush test is presented in Fig. 37.

The slope of the curve is roughly linear through a radial load of 400 pounds. The inverse

of the slope in this region corresponds to _,1. The value for cql from the crush test is

0.64 x 10-a inches per pound. This value is significantly smaller than the cq, value obtained from the in-plane static tests. Thus, the crush test exhibits a stiffer radial response. This discrepancy is likely due to the different boundary conditions. The

experimental boundary conditions for the crush test involved bolting the potted ends to

Experimental Results 78
Table 8. Iqexibility influence coefficients for channel specimen from out-of-plane tests

t_

3

eLateral Load 0[22 _33 0[23 0[32 0113 0[12

x 10 .2 x 10 l x 10 _ x 10 2 x 10 _

lbs in/lb 1/in-lb 1/lb 1/lb 1/lb in/lb

Q = 1.063 0.242 O. i 69 0.214 O. 120 O. 139 0.842

Q = 2.000 0.229 0.156 0.177 0.113 0.115 0.768

Q = 2.313 0.235 0.146 0.194 0.116 0.096 0.757

-.,.1

",0
1500

1200

900

i 600

300

0

0.00

......... i ......... ! ......... i ......... i ......... i .........

0.25 0.50 O. 75 1.00 1.25 !.50

RADIAL __ (W)

.-..-,, ORU_ TEST

Figure 37. Crush test load displacement plot: Load versus radial displacement for the crush test of

the l-section specimen.

Experimental Results 80
the I-beam where the static tests used clamps. It will be shown in the Chapter 6 that tile

in-plane response is very sensitive to small changes in the boundary conditions.

The crush test was conducted as part of a related research effort by Mr. E. Moas,

Graduate Research Assistant, and Professor O. H. Griffin that is sponsored by the

landing and Impact Dynamics Branch, NASA Langley Research Center. The failure and

post failure response of the frame will be addressed in detail by these researchers. The

failure of the frame will be presented here in an effort to be complete without benefit of

detailed analysis.

The initial Pailure event involved two simultaneous buckles in the web of the specimen. One buckle occurred at midspan and the other was located about eight inches to

one side. The buckles are shown in Fig. 38. The initial failure occurred at a radial load

of 1048 pounds. The strain in the region of the buckles at failure was significantly lower

than the anticipated material failure strain suggesting a local instability possibly

exacerbated by the unsynunetric layup in the web.

Experimental Results 81
I-"

t_

_'7-22

"C: _

Ci i-_:,

C)'

--f;

oo Figure 38. Initial failures for crush test: Local buckling in the web at the midspan and 10 degrees to one side were the first failures in the 1-section

crush tesL
Chapter 5

Analysis

Several structural analyses used to predict the linear static and vibrational response

of the curved composite test specimens are discussed in this chapter. The first section

traces the development of Vlasov's thin-walled, curved beam theory. One-dimensional

continuum solutions for the decoupled in-plane and out-of-plane vibrational, and the

in-plane static responses, are presented. Finally, the finite element computer program

developed by Noor et al. (Ref. 7) is discussed. The finite element is based on a Vlasovtype thin-walled curved beam theory and includes the additional effects of transverse

shear deformation and rotatory inertia. The finite element was developed specifically for

the analysis of curved thin-walled beams with open sections.

Analysis 83
Thin-Walled Curved Beam Theory

]he thin-walled curved beam theory presented here was originally developed by

Vlasov (Ref. 23). Culver (Ref. 22) used Vlasov's theory in his solution for the decoupled

out-of-plane vibrational response for a curved thin-walled beam with a doubly symmetric cross section. Culver included the inertia terms by treating them as a periodic distributed static load. In this development, the strain energy relations are formed similar

to Vlasov and then the kinetic energy is derived in a manner consistent with the

kinematic assumptions and includes the inertial effects due to warping displacements.

Hamilton's principle is used to obtain the differential equations of motion for a curved

beam with a single plane of symmetry coinciding with the plane of curvature. The theory is extended by incorporating the constitutive relations of laminated composites into

the t looke's law expression.

Strain Energy

A ring segment is defined with respect to cylindrical coordinates (r,0,y) as shown in

Fig. 39. The radius measured from the origin O is written as r -- a-z, where a is the radius

of the circle through the centroids of each cross section normal to the circumferential

direction, and z is a cross-sectional coordinate directed radially inward toward origin O.

Coordinate x is defined to be tile arc length on the centroidal circle (x= a0). At a generic

cross section, the orthogonal coordinates x, y, and z are defined by the right-hand-screw

rule. The cross section is assumed to be uniform along the circumference and to be

symmetric about the r-0 (or z-x) plane. Thus, the cross-sectional axes y and z are principal centroidal axes of the cross section.

Analysis 84
0

X

Z

Figure 39. Cylindrical and cross-sectional coordinates: The sketch shows a cylindrical coordinate

system for a curved beam with the origin 0 at the center of the beam and a cross-sectional

coordinate system (x,y,z) at some generic point on the arc.

Analysis

85
Relative to the cross-sectional coordinate axes y and z, shell coordinates s and { are

defined to describe a thin-walled element of the curved beam. A contour line C is defined that is at the middle of the wall thickness, and s is the arc length coordinate along

this contour. The contour is a continuous arc with a piecewise continuous tangent. The

tangent to the contour can jump at junctions betwcen tlanges and webs, for example.

The shell thickness coordinate { is measured normal to the tangent of the contour at s;

= 0 on the contour, and [_1_< h/2 , where h is the wall thickness. In general, h can be

a function of s. The positive directions of coordinates s and _ are fixed by defining

(_,s,0) as a right-handed orthogonal set of directions at a generic point in the shell space

as shown in Fig. 40. A material point in the wall of an element in the cross section located at s and _ is also described by coordinates y and z according to the relation

y(s, {) = y(s) + _ sin _(s) (5.1)

z(s, = - ¢cos (5.2)

in which _(s) and 2(s) are the coordinates of the contour C, and cc is the angle between

the positive y-axis and the positive tangent direction on the contour at s. Angle c_ is

positive if measured counterclockwise when viewed down the positive 0-axis. On the

contour, the differential coordinates satisfy

dy dg

ds - cos _ ds - sin c_ (5.3)

From Eqs. (5.1-5.3) the following derivatives are obtained for later use

0y Oz

Os - (1 + ¢/Rs) cos_ , Os - (1 + _/Rs) sins (5.4)

Analysis 86
8y 0z

0_ - sin_ , c_ - -cosec (5.5)

in which I/R, = da/ds, and R_ is the radius of curvature ofthe contour in the y-z plane.

The shear center for the section lies along the z-axis at z =7_,, because of the assumed symmetry. In thin-walled beam theory, the coordinates of a point on the contour

relative to the shear center are r(s), which is measured along the normal to the contour

at s, and q(s) which is measured along the tangent to the contour at s. Coordinates r and

q are called contour coordinates. Coordinate r(s) is not to be confused with tile cylindrical coordinate r introduced earlier. In general, the cylindrical coordinate r is replaced

with a-z. Thus, a material point in the wall located at s and _ can be defined in terms

of r, q, y, and z as follows

r(s) + _ = ysin_- (Z-Zp) COS (5.6)

q(s) = y cos _ + (z-Zp) sin (5.7)

The displacement components in the 0-, y-, and z- directions are designated u, v, and

w, respectively. In Vlasov's curved beam theory, these are given by

u(O,y,z) = U(O) -- y_z(O) + Zq_y(O) -- CO(S,_)"r(O) (5.8)

v(O,y,z) = V(O) - (z - Zp) _bx(O)

(5.9)

w(0,y,z) = W(0) + y_bx(0 ) (5.1o)

in which U(0) is the circumferential displacement of the centroid, V(0) and W(0) are

the y- and z- direction displacements, respectively, of the shear center, 4_x(0) is the rotation of the cross section about the shear center and is positive counterclockwise when

Analysis 87
z X'_'--_(s)

/I ,,,_,_c_ - -- _--_ '

_L. / .._ _\

g(s)\ I

\

'r\

x4-P, (O, Zp)

I

0

Figure 40. General symmetric cross section: The sketch illustrates the cross-sectional coordinates for

an open, thin-walled circular section. Contour coordinates: r(s) and q(s). Shell wall coordinates: s, _. Cartesian coordinates of contour: y(s), Z(s).

Analysis 88
viewed down the positive x-axis, _by(0) and _bz(0) are rotations about the y- and z- axes,

respectively, which are positive counterclockwise when viewed down their positive axes.

Except for the term o)r in the expression tbr u, the displacements in Eqs. (5.8-5.10)

represent a deformation for which plane sections remain plane. The term mr represents

warping of the cross section.

In cylindrical coordinates the linear strain-displacement equations are

eo0 = (U,o - w)/(a - z)

£yy ----- V,y

/_ZZ _ W'z

V0y = U,y + v,0/(a - z)

Y0z = U,z + (W,o + u)/(a - z)

)_yz = V,z nt- W,y

(5.11)

in which partial differentiation with respect to a coordinate is denoted by a conuna with

the coordinate following as a subscript. Substituting the displacements of Eqs. (5.8-5.10)

into the strains in Eqs. (5.11) results in

(a - z)Coo = U' - W - y(_b' z + _bx) + z_b'y - fo't" (5.12)

£yy _-- £zz = 0 (5.13)

(a - z)}'Oy = -(a - Zp)t_z -Jr V'-(z - Zp)(t_' x - t_z ) - (a - z)O),yT (5.14)

(a -- z)Y0z = W' + U + a_by + y(qS' x - _z) - [(a - z)co, z +co]r (5.15)

Vyz = 0 (5.16)

Analysis 89
in which a prime represents an ordinary derivative with respect to 0. Vanishing of the

strains in Eqs. (5.13) and (5.16) indicates there is no distortion within the cross section.

The rotation-displacement relations are taken as

q_y = -(W' + U)/a (5.17)

4_z = V'/(a - Zp) (5.18)

and the unit twist-rotation relation is taken as

z = (4/x - 4_z)/(a - Zp) (5.19)

As a consequence of Eqs. (5.17-5.19), the transverse shearing strains ?oy and Yozin Eqs.

(5.14) and (5.15) are rewritten as

(a - z)V0y = -[(a - Zp)(Z - Zp) + (a - z)rO,y]T (5.20)

(a - Z)yoz = [(a - Zp)y - (a - z)co, z - co]r (5.21)

The transverse shear strain components in the shell coordinates ({,s, 0) are Y0,and Yo¢.

They are given by the transformation equations

YOs = Y0y COS 0_ -_ Y0z sin cc (5.22)

Yo¢ = )'Oy sin u - _Oz cos c_ (5.23)

After substitution of Eqs. (5.20) and (5.21) into (5.22) and (5.23), the transverse shear

strains in shell coordinates are

Analysis 90
(a - z)_0s = [(a - Zp)(y sin cc - (z - Zp)cos Ix)

- (a - z)(_O,y cos Ix + o, z sin Ix) - 09 sin Ixlr

(5.24)

(a - z)y0¢ = [-(a - Zp)(y cos & + (z - Zp)sin Ix)

- (a - z)(o,y sin _ - co,z cos a) + o9 cos Ix]r

(5.25)

From the chain rule and Eqs. (5.4) and (5.5), partial derivatives of the warping function

in shell coordinates are

co,s = (1 + _/Rs) (O),y COS IX -t- fO,z sin IX) (5.26)

O),¢ _--- fO,y sin IX - co, z cos IX (5.27)

As a consequence of Eqs. (5.6) and (5.7) for the contour coordinates, and Eqs. (5.26) and

(5.27), the transverse shear strains are

(a - z)y0s = [(a - Zp)(r(s) + _) - (a - z)(1 + _/Rs)-log,s- o sin IX]r (5.28)

(a - z)yo; = 1-- (a - Zp)q(s) - (a - z)o),¢ + m cos _]r (5.29)

Using Eqs. (5.4) and (5.5) it can be shown that

(a - z)2 0 o

(I + _/Rs) c_s (a- z ) = (a- z)(l + _/Rylo,_ + _osinIX (5.30)

(a- z)2 4(a_m z) = (a-z)o),¢- o) cosIX (5.31)

and with these identities Eqs. (5.28) and (5.29) are rewritten as

Analysis 91
Zp)(r(s) + _) (l(a+- _/Rs) z)_ 0sa ( a _, - z )}T

(5.32)

(a - z)y0¢ -(a- Zp)q(s)- (a- , _( a--z T

(5.33)

The Vlasov assumption is that the transverse shear strain vanishes on the contour;

that is, yo_(s, _ = 0) = 0 . Let a_(s, _ = 0) = N(s), and the Vlasov assumption implies

from Eqn. (5.32) that

(a - Zp)r(s) - (a -- Z)2_Sd( a -- )_-0 (5.34)

Integrating Eqn. (5.34) with respect to s from s = 0 to s, we write

N(s) = (a - zp)(a - g(s))

s r(s) (a - _(s))2

ds (5.35)

in which the origin of s is selected such that thc integral in Eqn. (5.35) vanishes if the

integration limits are over the entire contour C. For thin rings, the largest cross-sectional

dimension is less than one-tenth the radius a. Thus, a-g _- a, and Eqn. (5.35) is approximated by

N(s) = r(s)ds (5.36)

which is the same expression as used in straight beam theory. The geometric interpretation of N in Eqn. (5.36) is twice the area swept-out by a ray whose one end is fixed at

Analysis 92
the shear center and the other end moves on the contour from s = 0 to s. For this reason,

the contour warping function of Eqn. (5.36) is called the sectorial area.

The Kirchhoff-Love assumption is appropriate for the thin-walled shell elements of

the curved beam. This implies the transverse shear strain component Yo¢vanishes. Setting the coefficient of z in Eqn. (5.33) to zero, and then integrating with respect to _,

we obtain

co(s, _) = E(a - Zp)q(s) q- (a - z)_(s)]/cos 0_ + _(s) (5.37)

in which b(s) is the thickness warping function. The first term on the right-hand-side

of Eqn. (5.37) is independent of _ and is identified as the contour warping function

N(s). As a result of this identification, the expression for the thickness warping function

is

(a - zp)

_(s) = (a - 2) q + R---_ (5.38)

where

I/R o = cos_/(a - 2) (5.39)

The quantity 1/Ro is the normal curvature of the 0-curve in the shell element reference

surface. Thus, the warping function for the cross section is

co(s,¢) = V(s) + g(s) (5.40)

with the contour warping function N(s) given in Eqn. (5.35) and the thickness warping

by Eqn. (5.38).

Analysis 93
Thedistribution of the shearstraincomponent_'o_throughthe thicknessof the shell

wall is approximatedasa linearfhnction of _.A seriesexpansionin _ of the coetticient

of the unit twist r in Eqn. (5.32), subject to the Vlasov assumption (5.34), Eqs. (5.2),

(5.4-5.7), and (5.38-5.40), results in

(a - Z)Yos = 2(a(a -- Zp) _) 2 { + O({2) } r (5.41)

The terms of order _2 and higher are neglected in Eqn. (5.41).

The nonzero strains are _oo, Eqn. (5.12), and Yosin Eqn. (5.41). These can be rewritten in the tbrm

(1 - 2]a)(1 + ¢/Ro)_00 = e - Y_:z + Z_Cy- a)(r'/a) (5.42)

(1 - gla)(1 + URo)Yos -

2(a - zp) 2

a(a - 7)

Cr (5.43)

where

(a - z) = a(1 - E/a)(1 + ¢/Ro) (5.44)

e = (u'- W)/a (5.45)

Kz = (_b'z + ¢'x)la (5.46)

Ky = _'y/a (5.47)

Eqn. (5.44) follows from Eqn. (5.2) and from the definition of l/Ro Eqn. (5.39). The

quantity e Eqn. (5.45) represents the circumferential stretching strain of the centroidal

line, Kz is the change in curvature out of the plane of the curved beam, and Ky is the

Analysis 94
change in curvature in the plane of the curved beam. Substituting Eqn. (5.1) for y, Eqn.

(5.2) for z, and Eqn. (5.40) for co, into Eqn. (5.42) gives the circumferential strain as an

explicit function of {. ]'he result is

(1 - g/a)(1 + _/Ro)%o = e - Y_z + ZK.y --

- ([sin_z +

(5.4s)

The strain energy increment for the curved beam is

- fvol(OOO&OO &U = + aOs&Yos)dVol (5.49)

in which aoo is the circumferential normal stress and ao, is the shear stress. The differential volume element is

dVol = (a - z)d0dA = (1 + ¢/Rs)(1 + _/Ro)d¢ (1 - g/a)dsad0 (5.50)

in which the area element within the 0-cross section is expressed in terms of shell coordinates s and _. The strain increments 6eoo and &Vo_in Eqn. (5.49) are obtained in terms

of the increments of the bar strains 6e, 6Kz, &_:y, and 6"r from Eqs. (5.43) and (5.48).

These strain increments are substituted into the strain energy increment (5.49) using the

volume element in Eqn. (5.50). Integrations over the cross section are performed, and

the resulting strain energy increment for the curved beam is written as

&U = [Nfe + My6tCy + Mz&x z + Mo,(6"r'/a) + Ts&r]ad0 (5.51)

in which 0 e (0_, 0,), 02 > 0,, and the curved beam resultants are defined by the contour

integrals

Analysis 95
N = fcNods (5.52)

My = fc(NOZ - M 0 cos cz)ds (5.53)

M z = - fc(NoY + M o sinc0ds (5.54)

Mo_ = _ fc(NON + Mo_)ds

(5.55)

T s = 2(1 - Zp/a)2fc

gos

(1 -

ds (5.56)

The shell resultants appearing in the contour integrals of Eqs. (5.52-5.56) are defined

by the following integrals through the thickness of the shell elements:

(No, Mo) = fh(1, ¢)aoo (1 + ¢/Rs)d¢ (5.57)

M°s = fl _a°s (1 + _/Rs)d _ (5.58)

Analysis 96
]'he curved beam resultant N acts in the circumferential direction at the centroid of the

cross section. The bending resultants My and Mz are vectorally directed along the positive y- and z-axes, respectively, by tile right-hand screw rule. The bimoment M,o is taken

to act in the circumferential direction at the shear center. The bimoment is statically

equivalent to zero force and zero moment. The Saint Venant torque T, is directed circumferentially and also acts at the shear center.

Kinetic Energy

]'he kinetic energy is

- l fv 92 T - 2 (62 + + _,r2)pdVol (5.59)

ol

where p denotes the mass density, and the overdot denotes a partial derivative with respect to time. Time derivatives of the displacements are determined from Eqs. (5.8-5.10).

Using Eqs. (5.1), (5.2), and (5.40), these time derivatives can be written explicitly in

terms of contour coordinates y(s), 2(s), N(s), and (5(s), and the thickness coordinate {.

The results are

fi(s, 0, {,t) = U(0,t) - y(S)_z(0,t ) + 2_y(0,t) - _#(0,t)

- _[-_z(0,t)sin _(s) + _y(0,t)cos a(s) + go(s)+(0,t)]

(5.60)

v(s, 0, _,t) = _r(0,t) -- (z(s) -- Zp)(_x(0,t ) + {6x(0,t) COS cz(s) (s.61)

_v(s, 0, {,t) = W(0,t) + y(S)&x(0, t) + _x(01t)sin _(s) (5.62)

Analysis 97
These relations for the velocity components are substituted into the kinetic energy (5.59)

along with tile volume element (5.50). Integration over the cross section is performed

and the result is

-T- 21 _°2_Toim_ad0 (5.63)

in which the 7xl velocity vector l) is

g(0,t) = [0, 9, W, _;x, _;y, _;z, '3'_ (5.64)

and the 7x7 symmetric mass matrix m is

o w

ml 1

Symm

0

m22

0 0

0 m24

n_133 m3a

n_4

by @z ":t"

m15 m16 m17

0 0 0

0 0 0

0 0 0

mss m56 m57

rY166 1T167

m77

(5.65)

Analysis 98
The nonzero elements of the mass matrix are the contour integrals

ln56 =

= m22 = na33 = J,.m(l -- g/a)ds 11111

ml5 = fc(rn_ - I¢ cos _)(1 - ?./a) ds

m16 = -m34 = - fc(mY + I¢ sin_)(l - g/a) ds

rill7 = - fc(mW+ i_)(1 - _,/a)ds

m24 = fc (-m(g - Zp) + 1¢cos c0(l - g/a)ds

m44 = fc(m(r2 + q2) + 2I¢r + I¢¢)(1 - g/a)ds

ms5 = _[m22 - 2I;gcosa + I¢¢ cos2cc](1 - g/a) ds

.[c[ -my--2 + l_(y cos a - g sin a) + I¢{ sin 0ccos a](l - g/a) ds

m57 = fc [ -mz---_ + I_( - g_ + cos ccN) + I¢_ cos c¢oJ](1 - g/a) ds

m66 = £[mY 2 + 2I¢_ since + I¢¢ sin2c_](l - g/a)ds

= fc[m + i¢(y_ + _ sin a) + I¢¢ sin cc_](1 - 2/a)ds

m77 = fc[mN2 + 2I_N_, + I¢_2](1 - g/a)ds

(5.66)

Analysis 99
in which the mass properties of the shell wall are

(m, I_, I_) = £(1, _, {2)p(l + _/Rs)(l + {/R0)d_

(5.67)

Hamilton's Principle

The variational statement of dynamic equilibrium is tIamilton's principle. For no

external loads acting on the curved beam, Hamilton's principle is

I t2(aT - 8U)dt = 0

t I

(5.68)

where the time limits satisfy ta > tl. The four independent functional degrees of freedom

are U, V, W, and _bx. Rotations qSy and q5z, and the unit twist "r, depend on on U, V,

W, and 4_x as shown by Eqs. (5.17-5.19). Lagrange multipliers are introduced to keep the

seven functional degrees of freedom U, V, W, 4_x, _by, _b,, and -c in the functional independent. The shear force Q, is the Lagrange multiplier associated with rotation _br, shear

force Qy is the Lagrange multiplier associated with rotation _bz, and the torque T is the

Lagrange multiplier associated with unit twist -r. The augmented functional is

(5.69)

Analysis 100
where

6U* = 6U +

ff{Qy[6V'/(a - Zp) - fi_bz] + Qz[(fiW' + 6U)/a + fiqSy]

+ q'[(6_b' x - 6_bz)/(a - Zp) - 6r]} adO

(5.70)

The variation of the kinetic energy obtained from Eqn. (5.63) and the variation of the

augmented strain energy in Eqn. (5.70) are substituted into the variational principle Eqn.

(5.69). In addition Eqs. (5.45-5.47) and (5.51) are substituted in the appropriate manner

into the variational principle. After these substitutions, the inertia terms are integrated

by parts with respect to time, and the variations in the degrees of freedom

6U, 6V, 6W, _3_bx,c_qS_,6_bz, and fir at times tl and t_ vanish according to Hamilton's

principle. Integrating by parts with respect to 0 yields the Euler-Lagrange equations associated with each functional degree of freedom.

6U: -mliLl - m,s_y- m16_z- m17_ + N'/a - Qz/a = 0 (5.71)

6V: -m22_ r - m24_× + Q'y/(a - Zp) = 0 (5.72)

6W: -m;3_/¢ - m34_ × + N/a + Q'z/a = 0 (5.73)

cSq_x : --m244Q -- mj_/¢ -- rrq4_x -- Mz/a + T'/(a - Zp) = 0 (5.74)

O_by: -mlsU - m55_y - m56_ z -- m57_ -4- M'y/a - Qz = 0 (5.75)

&hz : -ml6Ll- m56_y- m66@z - 1T167:_q- M'z/a

+ Qy -4- T/(a - Zp) = 0

(5.76)

6z: -m170 - msv_y - m67_z - in77:_ nt- M'oJa - T s + T = 0 (5.77)

Analysis 101
The seven boundary conditions at 0 = 01 and 0 = 02 are

Essential Natural

U N

V (1 - Zp/a)-lQy

W Qz

4)x (1 - Zp/a)-lT

t_y My

Cz Mz

z Moo

(5.7s)

Hookds Law

It is assumed that the shell elements comprising the thin-walled curved beam are

laminated from materials having monoclinic symmetry with respect to a {-surface (a

surface on which { is constant). Monoclinic symmetry is exhibited by an off-axis lamina

reinforced with continuous and aligned fibers. This is the case for the graphite-epoxy

specimens in this study. The fiber direction, cormnonly labeled the l-axis, is established

by a counterclockwise rotation through an angle 0 from the positive s-axis to the 1-axis,

looking down the positive { -axis. In beam theories, the lateral stresses are assumed to

vanish in IIooke's law. This assumption implies that stress components ass, a¢¢, and as¢

are zero in the generalized strain-stress relations. Thus, for an off-axis lamina the

strain-stress relations are

Analysis 102
_00 = 8'220"00 --J-- S'26O'0s

Yo¢ = S'a4ao¢

Yse = S'62aoo + S'66aOs

(5.79)

in which the off-axis compliances are

8'22 = Slln a + (2S12 + S66)m2n 2 + $22 m4

8'26 = 8'62 -__ (2Sll - 2S12 - S66)l-Dyl 3 - (2822 - 2S12 - S66)m3n

S'44 = $44 m2 + Sssn 2

S'66 = 2(2S11 + 2S22 - 4Sz2 - S66)m 2n2 + S66(m4 + n4)

(5.80)

with m = cos0 and n = sin O. The on-axis compliances in terms of engineering

constants are

$11 = 1/Ea, S21 = -v12/El, S22 = 1/E 2

S4a = 1/G23, S55 = 1/G13, 866 = I/G12

(5.81)

The transverse shear strain _o_ was assumed to vanish by the Kirchhoff-Love assumption. As a consequence, the second of Eqs. (5.79) is neglected. Writing the inverse of the

two remaining equations in Hooke's law from Eqs. (5.79) we have

aoo = R22e00 + R26YOs

O0s ----- R26goo + R66YOs

(5.82)

in which the reduced transformed stiffnesses are

(R22, R26, R66) = (8'66, -8'26, S'22)/1-S'22S'66 - (8'26) 2] (5.83)

Analysis 103
For an isotropic material, Ra = E, R26 = 0, R66 = G, where E and G are the modulus of

elasticity and the shear modulus, respectively.

The Hooke's law for the curved beam is obtained as follows. The strain relation for

_0o in Eqn. (5.48) and the relation for Yesin Eqn. (5.43) are substituted into Eqs. (5.82)

to get the stresses in terms of beam strains e, Ky, Kz, T'/a, and r. These stress-beam strain

equations are in turn substituted into the definitions of the shell resultants

No, Me, and Me_ in Eqs. (5.57) and (5.58). Finally, these shell rcsultant-bcam strain relations are substituted into the definitions of the beam resultants in Eqs. (5.52-5.56). The

result of these manipulations is

N1

My ]

vz I

_ Ts J

EA

Symm

ESy -ES z -ESo_ EH

Elyy-Elzy-Elcoy Ett c

EIzz EI_,z -EtI s

Elo_o_ -Et-lq

GJ

e

Ky

K z

"c'/a

T

(5.84)

in which

EA=

a22(l - g/a)-lds

(5.s5)

ESy = (a22z -- b22 cos c_)(l -- _/a)-lds (5.86)

Analysis 104
ESz = !(a22Y + b22 sin _)(1 - 2/a)-_ds

,0 (5.87)

ESco = £(a22_ q- b22_o)(1 - g/a)-Jds (5.88)

EH = 2(1 - Zp/a)2£b26(1 _ g/a)-2ds (5.89)

Elyy = fc(a22_2 _ 2b22gcos c_ + d22 cos2a)(1 _ g/a)-Ids (5.90)

Elzy = _c(a22_Y q- b22(z sin _ - y cos _) - d= sin a cos a)(1 - _/a)-lds (5.91)

EIc°Y = £(a22z'-_ + b22(z_o - _cos _) - d22_ cos _)(1 - k/a)-lds (5.91)

Analysis

105
_n

t_

c_

I

c_

$

J_

c_

÷

c_

c_ r_

÷

Jr

c_

o_

0

r_
GJ = 4(1 - Zp/a)afcd66(1 -- g/a)-3ds

(5.99)

In the contour integrals of Eqs. (5.85-5.99), which define the curved beam stiffnesses,

coefficients in the integrals appear that are based on integrated stiffness properties

through the shell wall thickness. The formulas for these integrated stiffness properties

are

(a22, b22, d22) = i(1, {, {2)R22(1 + {/Rs)(1 + {/Ro)-ld{ (5.100)

(a26, b26, d26) = £(1, _, _2)R26(1 + {/Rs)(1 + {/Ro)-ld_

(5.101)

d66 = _R66_2(1 + {/Rs)(l + {/Ro)-_d{

_h

(5.102)

Within the cross section, the positions of the modulus-weighted centroid, shear

center, and contour origin (position where s= 0), are determined by requiring selected

stiffness terms in Eqs. (5.84) to vanish. Requiring the modulus weighted first moments

ES_ and ESz to vanish locates the z and y positions, respectively, of the centroid. The y

and z positions of the shear center are determined by setting the modulus-weighted first

Analysis 107
sectorial moments EI.,y and EI.,z, respectively, to zero. The condition that the modulusweighted sectorial area ES.. vanishes determines the position of the contour origin.

Finally, the direction of the principal axes is determined from the condition that the

modulus weighted product moment EI,y equals zero. Thus, ttooke's law for a curved

beam using principal coordinates of the cross section simplifies to

EA 0 0

Elyy 0

Elzz

Symm

0 EH

0 -EI-I c

0 -EH s

E Icoco -EHq

GJ

(5.103)

The stilTness matrix in Eqn. (5.103) is not diagonal because of the presence of the "EH"

terms. If the laminated wall construction is specially orthotropic, then the terms

Eli = El-It= El-I,= EIIq= 0 , and tile stiffness matrix is diagonal. For a specially

orthotropic laminate, the lamina fiber angles (9) are either zero or ninety degrees so

S'26 = 0 in the second of Eqs. (5.80), and R=6 = 0 in Eqs. (5.83). For R=6 = 0, Eqn. (5.101)

shows that shell stiffnesses a26= b,6=d26= 0, and consequently EH, Et{c,

EI-t, and EHq all vanish.

For a symmetric laminated wall construction, shell stiffness b a in Eqn. (5.100) and

b26 in Eqn. (5.101) are zero. This simplifies the computation of the beam stiffnesses in

Eqs. (5.85-5.99). The term EH in Eqn. (5.89) is zero if b,6 is zero, but terms

EI-I_, EI-I,, and EI-Iq are not zero because d26 is not zero. The shell stiffness term d26 reflects bend-twist coupling of symmetric laminates containing off-axis plies. For straight

beams with syrmnetric laminated walls, Bauld and Tzeng (Ref. 27) have derived similar

terms to the Eli terms presented here. These authors point out that Vlasov's assumption

that Y0s= 0 on the reterence surface is only applicable to symmetric laminated wall construction. For thin curved beams, the largest cross-sectional dimension is less than

Anal3sis 108
one-tenth the radius a. It is permissible, then, to neglect g(s)/a and zp/a with respect to

unity in all the tbrmulas presented for the curved beam. For thin walls h/R_ and h/Re

are also small with respect to unity, and are neglected in the shell wall stiffness formulas

of Eqs. (5.100-5.102). Conscquently, for thin curved beams with thin wall cross sections,

the formulas for the beam stifthesses, Eqn. (5.85-5.99), reduce to those of a straight

beam theory.

If the thin-walled curved beam is made of an isotropic and homogeneous material,

then the reduced stiffnesses are R22 = E, R26 = 0, and R66 = G. Using principal coordinates in the cross section, and assuming a thin curved beam with thin walls made from

an isotropic and homogeneous material, Hooke's law is

N = EAe

My = I_IyyKy

M z = EIzz_Cz

Mo_ = EI_o_z'/a

T s = G Jr

(5.104)

In Eqs. (5.104), E is Young's modulus, G is the shear modulus, A is the area of the cross

section, I_ is the second area moment about the y-axis, I_ is the second area moment

about the z-axis, Io,_,is the second sectorial moment (or the warping coefficient), and J

is the effective polar moment for Saint Venant's torque T,. These geometric properties

of the cross section are given by the formulas

Analysis 109
Iyy -_-

Izz

l e.o oo _---

A = fchdS

c[hZ2 + (h3/12)cos2e]ds

c[hY 2 + (h3/12) sin2_]ds

c[h_2 + (h3/12)_o2]ds

J - 3 h3ds

"C

(5. o5)

In some thin wall beam theories the terms with h a in Eqs. (5.105) are neglected except

in the equation for J.

Continuum Solutions

This section describes the development of continuum solutions for the in-plane

static and vibrational response for a thin curved beam with homogeneous, isotropic,

thin-walled, construction. For the special case of a symmetric cross section, the

equations decouple allowing individual solutions for the in-plane and out-of-plane responses. An out-of-plane vibration solution is also presented.

Analysis

110
In-Plane Vibration Solution

The equations of motion as stated in Eqs. (5.71)-(5.77) are extensively coupled by

the inertia terms. A continuum solution for the coupled equations would be very difficult. For our special case, the differential equations of motion which govern the the

in-plane response (5.71, 5.73, and 5.75) reduce to

6U: -m110 + N'/a - Qz/a = 0 (5.106)

6W: -m33_9 + N/a + Q'z/a = 0 (5.107)

6qby: -m55qSy + M y/a - Qz = 0 (5.108)

in which m n =-m33 = pA = _a, and nhs = plr_ • Solving Eqn. (5.108) for Qz and substituting into Eqs. (5.106) and (5.107) yields

-n0 + N'/a = -plyy_y/a -4- M'y/a 2 (5.109)

--_'v/¢ + N/a = plyy_y/a 2 - M"y/a 2 (5.110)

Using the expressions for 42y in Eqn. (5.17), My and N in Eqs. (5.104), e in Eqn. (5.45),

_y in Eqn. (5.47) and substituting into Eqs. (5.109) and (5.110) yields

pIyy ..",

EA (U" EIyy (W'" + U") = NU + ----_(w + U) (5.111) ---5- -w')+ ----ra a a

EA Elyy (W .... + U'") = _,_r pIyy (_vV" + U;') (5.112) 2 (U' - W) 4 2

a a a

Analysis 111
A separable solution is assumed of the form

W(0,t).)

(5.113)

in which i= x/-_-, ,t is the frequency in radians per second, and _/ is an unknown parameter. To determine _/, first substitute Eqn. (5.113) into Eqs. (5.111) and (5.112) to

get

(5.114)

where

C1 EA Elzz m55 1

= _/2[--T- + a ] + 22[_ + 2 J

a a a

-- Elzz EA 22 mss C_ = r/[r/2 4 2 + ---5-]

a a a

-- 4 Elzz EA 22[N _ r/2 m55 1 C4 = ;7 a + 2 2

a a a

(5.115)

Second, a nontrivial solution for A and B in Eqs. (5.114) requires that the determinant

of the coefficient matrix vanishes, and this leads to a cubic equation (characteristic

equation) in r/a. The six roots of _/are denoted _/j,j = 1,2, ... 6, and may occur as complex

conjugate pairs. For each root G, Eqs. (5.114) also yield the eigenvector components

Pj = (B/A)j , t/= t/j (5.116)

Thus, the general solution to Eqs. (5.111) and (5.112) is

Analysis 112
6

+ Z{lt U(0,t) = e i}'t Cj e 'Tj°

w(0,t)) I,j

j=l

(5.117)

where Cj are the unknown constants. The constants Cj are determined from the boundary conditions. For example, clamped end conditions, see Eqs. (5.78), require U, W, and

qSy to vanish at 0 = 0_ and 0 = 02. This leads to the six homogeneous equations

6

U(01) = ECj e '?p_ = 0

j=l

6

U(02) = ECj e '_p2 = 0

j=l

6

W(01) = ECj Pj e_j°_ =0

j=l

6

W(02) = ECj Pje '7j02 = 0

j=l

6

W'(01) = ECj Pjr/j e_j°l = 0

j=l

6

W'(02) = ECj Pj_/j e 'h°_ = 0

j=l

(5.118)

These are written in matrix form as

A C = 0 (5.119)

in which A is a 6 x 6 matrix and CT = [Cj] . Nontrivial solutions to Eqn. (5.119) require

the determinant of A to vanish. Since the six roots oft/are functions of 2, and t/appears

as an exponent in the coefficients of matrix A, this is a nonlinear eigenproblem for the

Analysis 113
eigenfrequency 2 . The solution proceeds by a determinant plotting method, in which 2

is assumed, the six roots of t/j are computed along with the six values of l'_, and the determinant of A in Eqn. (5.119) is calculated. Increasing values of 2 arc selected until a

sign change in the determinant occurs, and Newton's method is used to converge on the

frequency. For a given frequency, the eigenvectors C of Eqn. (5.119) are obtained to

give the mode shape. Although the deternfinant plotting technique is a not very good

numerical method for large matrices, it works well enough for the matrices encountered

in this problem.

The boundary conditions for the free-free case are

N(O0 = N(09 = o

My(01) = My(02) = 0

Qz(0,) = Qz(02) --= 0

(5.120)

The equations for pinned boundary conditions are

u(o,) = u(o2) = o

W(Ol) = W(02) = 0

My(01) = My(02) = 0

(5.121)

The equations for pinned boundary conditions with torsional springs are

u(ol) = u(o2) = o

w(ol) = w(o2) = o

My(01) = OCTW'(01)

My(02) = 0_TW'(01)

(5.122)

Where c_T is the torsional spring constant.

Analysis 114
In-Plane Static Solution

For a thin curved beam with isotropic, homogeneous, thin-walled construction and

a monosymmetric cross section, the equilibrium equations describing the in-plane static

response can be obtained from the equations of motion (5.106-5.108) by discarding the

inertia terms. The resulting equations are

5U: N'/a - Qz/a = 0 (5.123)

5W: N/a + Q'z/a = 0 (5.124)

3_by: M'y/a - Qz = 0 (5.125)

Eliminating Qz in Eqs. (5.123) and (5.124) using (5.125) yields

N'/a - M'y/a 2 = 0 (5.126)

N/a + M"y/a 2 = 0 (5.127)

Substituting the relations fbr N and My from I Iooke's law in Eqn. (5. I04), Eqs. (5.126)

and (5.127) become

EA Elyy

(U"- W') + (W'" + U") = 0 (5.128) 2 4

a a

EA Elyy (W .... + U"') = 0 (5.129) 2 (u'- w) 4

a a

where A and I, are defined in Eqn. (5.105). Dividing Eqs. (5.128) and (5.129) by

EA/a 2 results in

Analysis i 15
u" - w' + t2(w'" + u") = 0 (5.130)

u'-w-t2(w .... + u'")=o (5.131)

in which 32 = Irj/(Aa 2) is the square of the slenderness ratio.

Assume the in-plane displacements can be represented by

{w}= {i}e _70

W

(5.132)

Substituting Eqs. (5.132) into the differential Eqs. (5.130) and (5.131) leads to the following characteristic equation for _/

_2t/2(_/2 + 1) 2 = 0 (5.133)

The roots of this characteristic equation are 0, 0, + i, + i, -i, -i, in which i = ,f-2-i-. Thus,

there are three repeated roots for _I, and the procedure presented by Kaplan (Ref. 32)

was used to construct six linearly independent solutions to Eqs. (5.130) and (5.131). The

solution is

c3_COS0_ + (sin

{;}---= Ct{I}-+-C2{01} q- {--sin0) C4lcos2}

+ C5 {0cos0}{0sin0}+ -0sin0 + 7cos0 C6 0cos0 - rsin0

(5.137)

where _ is defined as l-I - }2]/[1 + }2].

The solution in Eqn. (5.137) satisfies the differential equilibrium equations exactly

and can be used to model the static response of a thin curved beam subjected to combined in-plane loading. A curved beam subjected to a radial load f,, a tangential load re,

Analysis 116
and an in-plane couple whose moment is mr at a location defined by 0 = 0' is shown in

Fig. 41. The curved beam is supported at 0 -- (0,02) and 0z < 0' < 02.

This loading requires that two solutions be implemented over the beam, one for the

left portion (0_ < 0 < 0') of the curved beam and one for the right portion (0' < 0 < 05).

Thus, the solution for the left-hand-side is

6

= 2Cj_j(0) 01 < 0 < o. Wt.

j=l

and for the right-hand-side

(5.138)

6

= <0<0 2

WR

j=l

(5.139)

where 6j(0) are the six linearly independent solutions from Eqn. (5.137).

The problem requires twelve equations to solve for the twelve unknown constants

Cj. Six of the equations can be obtained from transition conditions at 0". Continuity of

the beam at 0" requires that the displacements and rotation at 0' are continuous requiring

u(o:_)= u(o__)

w(o'_.) = w(o+)

w'(o*) = w '(o+) *

(5.14o)

Analysis 117
W

U

Figure 4 I. Loading for in-plane static solution: A curved beam is subjected to radial, tangential, and

bending loads at 0 = 0". The curved beam is supported at the ends

(0 = 0land0 = 02).

Analysis 118
in which a quantity with a "+" superscript is evaluated from the right interval and a ....

superscript implies the quantity is evaluated from the left interval.

Equilibrium of a infinitesimal element at 0 = 0" provides three jump conditions in

the actions given by

N(O+)- N(O'_)= - ro

M(0+) - M(02) = - my

Qz0;_ - ez(0-) = l°r

(5.141)

The remaining six equations can be obtained from the boundary conditions at

0 = 0, and 0 = 02. The boundary conditions for clamped, pinned, free, and pinned with

a torsional spring are provided in Eqs. (5.118) and (5.120-5.122), respectively. Thus,

using three boundary conditions at 0 = 0_, three continuity conditions of Eqs. (5.140),

the three jump conditions of Eqs. (5.141), and three boundary conditions at 0 = 02 , a

set of linear equations can be written in the form

A C = F (.5.142)

where A is a 12 x 12 matrix of coefficients, C is a 12 x 1 vector containing the unknown

constants, and F is a 12 x 1 load vector. The solution vector C was determined by

Gaussian elimination.

Having found the solution vector, the displacements of the beam are given by Eqs.

(5.137-5.139). Differentiating and substituting into Eqs. (5.45) and (5.47) gives the extensional strain and the change in curvature, respectively. Substituting the strains into

Eqn. (5.104) provides the beam actions N and My.

Analysis 119
Out-of-Plane Vibration Solution

The differential equations of motion which govern the out-of-plane vibrational response of a thin curved beam with thin, isotropic, homogeneous, walls can be obtained

from Eqs. (5.72), (5.74), and (5.76) and (5.77). For the special case of a symmetric cross

section using principal centroidal axes, these equations reduce to

gV: -m22Q - m24_x + Q'y/a = 0 (5.143)

6¢x: -m249 - rn44_ x - Mz/a + T'/a = 0 (5.144)

64)z : -m66_z + M'z/a + Qy + T/a = 0 (5.145)

_Sr: -m77:_ + M'o,/a - T_ + T = 0 (5.146)

Having assumed the cross-sectional coordinates are the principal centroidal axes,

m2, = m, lzp = Nzp, m,, = p[(I_ + I,z) + Az_] , m66 = plz,, and m77 = plo_., where A,

Iyy, Izz, I_., are defined in Eqs. (5.105). If the cross section is doubly symmetric, the

centroid coincides with the shear center and zp is zero. Solving Eqs. (5.145) and (5.146)

for the shear force Qy and the torque T, respectively, and substituting into Eqs. (5.143)

and (5.144) yields

tt

T' M _o_o M"z Plo,,o ¥, s +

2 3 2 2

a a a a

PIzz

a _'z + _Q + i--nZp_x(5.147)

T' s M"_, M z -PI_°" _" + rn44_x + NZp9 (5.148) a 2 a a

a

Analysis 120
Using Itooke's law Eqn. (5.104) and substituting for _bz,r, and K, using Eqs. (5.18),

(5.19), and (5.46), respectively, Eqs. (5.147) and (5.148) become

-_9 + -_Zp_x plzz 9" + --(i_x" pI_,,_, - ---g-) 9" + _ (G" V" )

2 3 a

a a a

(5.149/

+ _( 4_×" + V .... EI,,,_, .... V .... a ) 5 (Chx a ) = 0

a a

_Zp9 + _4_x ;,I_ .. ,, 9- GJ ,, W

2 (G a ) 2(G a )

a a

Elzz V" EIo,o_ V""

+ 2 (G+ --7)+ 4 (G .... a

a a

)=0

(5.15o)

A separable solution is assumed of the form

G(0, t))

(5.151)

in which i = x/-L-I-, ,t is the natural frequency in radians per second, and r/is an unknown

parameter. To determine _/, first substitute Eqs. (5.151) into Eqs. (5.149) and (5.150) to

get

(5.152)

where

_._ El 2 GJ

a a a

+ ,_2[__ + _/2(Plzz + PI_o_, )q 2 4 --

a a

C2 = -7'/4 EI°'_° 2r GJ

a5 + _/ L--_ +

Elzz Plo,_,

3 ] + ,_2[__3Zp 3 _23

a a

(5.153)

Analysis i 2 !
-- 4 El_,o_ 2 GJ Elzz PI_,o_

C4 = _/ 4. r/ ---7- + 2 + 3-2[-m44 + 2 t/2]

a a a a

A nontrivial solution for A and B in Eqs. (5.152) requires that the determinant of the

coefficient matrix vanishes, and this leads to a quartic equation (characteristic equation)

in _/2. The eight roots of _/ are denoted rh , j = 1,2, ... 8, and may occur in complex conjugate pairs. For each root _/j, Eqs (5.152) also yield the eigenvector components

pj = (B/A)i , r/= r/j (5.154)

Thus, the general solution to Eqs. (5.149) and (5.150) is

8

 v,0 {dPx(O,t,}: t) e ? cj {1} Pj

j=l

e '_fl (5.155)

where Cj, j -- 1,2, ... 8, are unknown constants. ]'he constants Cj are determined from the

boundary conditions. For example, clamped end conditions, see Eqs. (5.78), require

V, _b_,V', and qS', to vanish at 0 = 01 and 0 = 02. This leads to the eight homogeneous

equations

Analysis 122
8

V(01 ) = 2Cj e'h°l = 0

j=l

8

V(02) = _Cj e'_j°_= 0

j=l

8

4_x(0_) = _-'Cj Pi e'TJ°_= 0

j=l

8

qSx(02) = ZCj pj e'TJ°_ = 0

j=l

8

V,(01) = ,__jCi r/ie 'Tj°j = 0

j=l

8

v I,U2) = LCj r/j e rli02 = 0

1=1

8

4/x(0 0 = 2Cj Pj rtj e'Tj°l = 0

j=l

8

_'x(02) = 2Cj Pj nj e_j°_ = 0

j=l

(5.156)

These are written in matrix form as

A c = o (5.1._7)

in which A is an 8x8 matrix and _CT= [Ci]. Nontrivial solutions to Eqs. (5.152) require

the determinant of A to vanish. Since the roots of t/are functions of,t , and r/ appears

as an exponent in the coefficients of matrix A, this is a nonlinear eigenproblem for the

eigenfrequency ,1. The solution proceeds by a determinant plotting method, in which ,1

is assumed, the eight roots rh are computed along with the eight values of Pj, and the

determinant of A in Eqn. (5.152) is calculated. Increasing values of 2 are selected until

a sign change in the determinant occurs, and Newton's method is used to converge on

Analysis 123
the frequency. For a given frequency, the eigenvectors C of Eqn.

to give the mode shape.

The boundary conditions for the flee-free case are

(5.155) are obtained

Qy(Ol) = Qy(O2)= o

T(01) = T(02) = 0

Mz(0_) = Mz(02) = 0

(5.158)

The equations for pinned boundary conditions are

,,r(n ?, - 1Z(A._ _ 0.

G(o_) = G(o_) = o

Mz(01) = Mz(02) = 0

M,o(O,) = Mo_(02) = 0

(5.159)

Finite Element Computer Program

The finite element computer program used extensively in this study was developed

by Noor et al., (Ref. 7) _.t?g.qifi_olly for the flee-vibrational analysis of curved thin-walled

beams with open sections. The analytical formulation is based on a Vlasov-type, thinwalled, curved beam theory similar t.o the theory discussed in this chapter. This section

briefly discusses the analytical development of the finite element relying heavily on the

equations presented previously to point out some interesting differences. Some practical

aspects of using the program will also be related.

Analysis 124

• . . -..
8

V(01) = ECj e 'TA = 0

j=l

8

V(02) = ECj e '7'°' = 0

j=l

8

_bx(01) = _-]Cj Pje '7'°_ = 0

j=1

8

q5×(02) = _--]Cj Pj e_j°_ = 0

j=l

8

V'(01) = _-_Cj r/j e_i°L = 0

j=l

8

V'(02) = )__Cj r/j e_j°2 = 0

j=l

8

q_'X(01) = _-_Cj Pj r/j e '7i0' = 0

j=l

8

q_'x(02) = _-]Cj Pj r/j e '7'02 = 0

j=l

(s.156)

These are written in matrix form as

A C = 0 f5.157)

in which A is an 8x8 matrix and G r = [Cj]. Nontrivial solutions to Eqs. (5.152) require

the determinant of A to vanish. Since the roots oft/are functions of)., and r/ appears

as an exponent in the coefficients of matrix A, this is a nonlinear eigenproblem for the

eigenfrequency ).. The solution proceeds by a determinant plotting method, in which 2

is assumed, the eight roots _lj are computed along with the eight values of Pj, and the

determinant of A in Eqn. (5.152) is calculated. Increasing values of 3. are selected until

a sign change in the determinant occurs, and Newton's method is used to converge on

Analysis 123
the frequency. For a given frequency, the eigenvectors C of Eqn.

to give the mode shape.

The boundary conditions for the free-free case are

Qy(Ol) = Qy(O2) = o

T(01) = T(02) = 0

Md00 = Md02) = 0

The equations for pinned boundary conditions are

(5.155) are obtained

(5.158)

v(o,) = y(q=) = o

4x(O_) = 6x(O2)= o"

Mz(01) = Mz(02) = 0

Moo(01) = Moo(02)= 0

(5.159)

Finite Element Computer Program

The finite element computer program used extensively in this study was developed

by Noor et al., (Ref. 7) _p.e:cifi.cally for the free-vibrational analysis of curved thin-walled

beams with open sections. The analytical formulation is based on a Vlasov-type, thinwalled, curved beam theory similar to the theory discussed in this chapter. This section

briefly discusses the analytical development of the finite element relying heavily on the

equations presented previously to point out some interesting differences. Some practical

aspects of using the program will also be related.

Analysis 124
Element Formulation

The finite element developed by Noor et al., includes the effects of transverse shear

deformation which were not included by Vlasov nor in the development presented in this

chapter. A modified form of the Hellinger-Reissner mixed variational principle was used

to develop the element. Including transverse shear deformation adds an additional two

degrees of freedom, and reduces the continuity requirements from C 1 to C O for the

transverse displacements V and W.

The Hooke's law relation used in the element development is

X

Mz

My

Mo_

Qy

Qz

Ts

EA

SYMM

0 0 -ES_o 0 0 0

EIzz -Elzy EIo_z 0 0 0

EIyy -Elo_y 0 0 0

EI_,_, 0 0 0

GAy 0 0

GA z 0

GJ

e

K z

Ky

kv (5.160)

0

Y0y

0

Y0z

in which the additional curved beam strain measures relative to those presented earlier

are

xiJ= _b',x/a 2

o = V'/a- Cz YOy

Y0z o = (W' + U)/a + C_y

(5.161)

The notation used by Noor has been transliterated into the notation used in this study.

The finite element code was developed for isotropic homogeneous materials, thus, the

Analysis 125
"Ett" stiffness terms characteristic of composite materials do not appear. A consequence

of this is that the finite element model cannot model the bend-twist coupling of laminated specimens.

The coordinate system used in calculating the section properties places the origin

and the pole at the centroid; thus, only the first moments of area Sy and S, are zero. A

diagonal stiffiaess matrix offers tEw advantages in a numerical solution. The section area

properties Ay and A, provide the opportunity to model the shear stiffness employing

shear correction factors to the shear stiffness GA.

The torsional strain component _: in Eqn. (5.160) is consistent with the torsional

strain used in the tlooke's law relation in Eqn. (5.104). However, the warping strain

component _I' in the first of Eqs. (5.161) is not the same as the strain measure z'/a used

for the bimornent in the Hooke's law of Eqs. (5.104).

To achieve C O continuity for the rotation q_x a new kinematic variable 0 ° is introduced that is deiined to be the derivative of rotation _b_. This kinematic relation is enforced via a Lagrange multiplier A. Thus, both 0 ° and _bx are represented by Co

interpolation functions in the element.

The functional used in the element development is

FI = FIHR + A(0'x - 0°) ad0 2Ep (A)2ad0

"C _'C

(5.162)

where I-I_R is the Hellinger-Reissner functional, A is the Lagrange multiplier associated

with the constraint condition, and ep is the penalty parameter in the regularization term.

The finite element equations for each element can be cast in the following compact

form

Analysis 126
[Fosji 0 -_p _- _2 [ooo] 000

St Q 0 0 0 M X

(5.163)

where {H}, {,/}, and {X} are the internal force parameters, Lagrange multiplier parameters, and nodal displacements, respectively; [F] is the flexibility matrix, [S] is the straindisplacement matrix. Finally, [P] and [Q] are matrices associated with the constraint

condition and the regularization term in the augmented functional.

The regularization term in Eqn. (5.162) results in replacing one of the zero diagonal

submatrices in the discrete equations (Eqn. 5.163) of the Lagrange multiplier approach

with a nonzero diagonal matrix, [P]/r.p. Mathematically, the submatrix cannot be zero

but physically it should be. The penalty parameter is used to keep the regularization

term small. Thus, an important consideration in this formulation is the proper selection

of the penalty parameter. The accuracy of the solution increases with increasing values

of tp. ltowever, for very large values, the equations become ill-conditioned, thereby increasing the round-off errors.

Application to Test Specimens

Noor's finite element program was developed for vibrational analysis, but it can be

used to predict the static response as well, thus, it was used to model both the static and

the vibrational tests. In the static mode, loads are applied to the model at the nodes and

the program predicts the deformations and strains caused by the applied loads. When

the program is run in the dynamic mode, it calculates the ten lowest natural frequencies

(eigenvalues) and their associated mode shapes (eigenvectors). In addition, the program

Analysis ! 27
g -j

calculates the strain eDergy distribution in each mode.. l'l_e strAn energies due to axial

strain, shear strain, in-plane bending, out-of-plane bending, torsion, and warping are

calculated for each mode.

The strain energy distributions for the modes of the I-section and channel section

specimens are presented in Fig. 42. The extensional, shear, and warping energies are

uniformly small in all modes. The strain energies in the I-specimen modes are either

pure in-plane bending or a combination of torsional and out-of-plane bending energies.

Thus, the modes of the l-specimen were easily identified as in-plane or out-of-plane

modes on the basis of strain energy distribution.

The strain energy distributions associated with the modes of the channel specimen

always exhibit some degree of coupling between the in-plane and out-of-plane responses.

The channel specimen's even numbered modes are dominated by in-plane bending energy and the remaining modes are dominated by torsional and out-of-plane bending energies. In general, if the energy in a mode consists of more than 85 % in-plane bending,

than that mode is considered an in-plane mode. If out-of-plane bending and torsional

energies comprise more than 85 % of the total energy, than that mode is considered an

out-of-plane mode. All of the modes of the channel specimen fit one definition or the

other, thus, the modes are easily classified based on their energy distribution.

With any finite element code, it is necessary to determine the number of elements

required for convergence. Convergence studies for static and dynamic cases were conducted for both test specimens. The results of the dynamic convergence study are presented in Fig. 43. The natural frequencies converged for a sixteen element model. The

sixteen element model was selected for both the static and dynamic analysis of both

specimens from this convergence study. The elements have three nodes each providing

thirty-three nodes and 231 degrees of freedom (7 degrees of freedom per node).

Analysis 128
o

1 I $ 4 I 8 7 8 0 I

_ W_ _ _ I____J

(a)I-_ECL_[N

!oo

0

1 I $ 4 | I 7 O 0 I

Id(X)E

WARPH3 m SHEAR

(B) CHAI_k-u, SPECDdEN

Figure 42. Strain energy distributions in vibrational modes for both specimens: The strain energy

distributions in the vibrational modes of both specimens are presented. Modes are arranged in order of increasing frequency.

Analysis 129
soo!

i

J

2ooj

i

123 123 123 123 123 123 123 123 122

I 2 _ • 5 _; 7 3 9

sooI

i

i

_23 _ NUkCOB_

tO X

"°i

I 2 5 LL 5 8 7 8 g 10 X

l

'm32

(B) CHANNEL SPECIMEN

Figure 43. Number of elements convergence study: The figure shows the convergence of the natural

frequencies of both specimens for the 8, 16, and 32 element models. Modes are in order

of increasing frequency. Natural frequencies are in hertz.

Analysis 130
An important consideration in using the finite element program is the proper selection of the penalty parameter. The penalty parameter was chosen based on a convergence study conducted for static and dynamic cases for both specimens. The penalty

parameter convergence studies were run for the sixteen element models. The convergence studies for the dynamic case for both specimens are presented in Fig. 44. The

plots show that above a certain value of the penalty parameter the natural frequencies

are constant. The smallest value of the penalty parameter for which all the predictions

changed by less than 1% was the value used in the subsequent analyses.

The plots show the affect of the penalty parameter on the natural frequencies of the

second, third, and tburth out-of-plane mode and the first in-plane mode for both specimens. The I-specimen plot is smooth and continuous while the plot for the channel

specimen is discontinuous. This is because the penalty parameter had no affect on the

energy distributions in the I-specimen modes. Tlms, the character of the mode was

constant and only the natural frequency varied. Further, the penalty parameter only affected the out-of-plane modes of the I-specimen. Neither of these statements are true for

the channel specimen.

The energy distributions in the channel modes varied extensively with the penalty

parameter. A mode which was initially dominated by in-plane energy might eventually

evolve into an out-of-plane mode, and an out-of-plane mode might gain in-plane energy

until it had to be considered an in-plane mode. Thus, the penalty parameter affected

both in-plane and out-of-plane modes. For penalty parameters in excess of 1 x 109, both

the natural frequencies and energy distributions of both specimens had converged. The

static and dynamic response of both specimens converged at the same value of the penalty parameter, thus, 1 x 109 was used as the value for the penalty parameter in subsequent analyses.

Analysis 131
IS0

g

100?5

0

o

,+oj

1251

Figure 44.

3--g--_-:_--g.

\

"_'" _-- C-- <:-- G-- -G- C----G--

._.a._e __..B.___.a_._ =___.__g_.g_:g--_:__=..--_.-_

¢

.... , .... , _ ,, .... , ..... , .....

3 6 9 12

.-.-. 214) OUT-OF-R.N_ _° 1ST

(A) I-SPECIMEN

IS

c----_----_-__._ /-c"--...c__._ _ ....

-t" _ _ -I- _..._.. ,_, ._.. _"l- ..... --_ ......

13...... e-- ..... -f3...... _.......

0

o _.......... ,;......... ,_ ; io

LOG R_iN.TY PN_lkLrn_ ffl)

----. 2ND OUT-OF-R.AN[ .+- 1ST

+-','-,'3flD OUT_ ++-, 4TH OUT"OF'R.N4E

(B) C_ SPECIMEN

Penalty parameter convergence study: The plots show the effects ofthe penalty parameter

on the natural frequencies of the second, third, and fourth out-of-plane modes and the first

in-plane mode for both specimens. The log of the penalty parameter is plotted on the

horizontal axis.

Analysis 132
The section properties used in Hooke's law in Eqn. (5.160) were also used in the

mass matrix. These section properties were evaluated using a computer program presented by Coyette (Re['. 33). This program calculates the section properties for a thinwalled open cross section assuming the branches of the cross section are straight and are

made of an isotropic homogeneous material. This program was subsequently modified

to evaluate the modulus-weighted section properties in Eqs. (5.85-5.99) assuming the

layup in the branches is symmetric. The modulus-weighted section properties were used

in the static analysis. Since the finite element program uses the input section properties

in both the stitIi_ess and the mass matrices, the modulus-weighted section properties

could not be used in the vibrational analysis. A listing of the section properties of both

specimens is presented in Appendix C.

Analysis 133
Chapter 6

Correlation of Numerical and Experimental Results

The numerical results from the finite clement solution and the continuum solutions

are presented with the corresponding experimental data for the I-specimen and channel

specimen. In the first section, natural frequencies and mode shapes from the analyses

are compared with the experimental data from the free-free and tile clamped-clamped

frame tests. In the second section, the predicted flexibility influence coetficients are

compared with coefficients calculated from the static test data. The comparisons of primary interest involve the predictions from the finite element solution and the experimental data.

Correlation of Dynamic Data

The finite element program was used to predict the three-dimensional vibrational

response of both specimens for clamped and pinned boundary conditions. The in-plane

Correlation of Numerical and Experimental Results 134
continuum vibrational solution was used to model the flee-free frame tests of the Ispecimen. The finite element program could not model the free-free frame tests because

of the presence of the rigid body modes. ]'he continuum solution also provided the

versatility to model the end conditions as torsional springs providing valuable insight

into the physical problem. The correlation of the free-flee vibrational data is presented

first followed by the correlation of the clamped-clamped vibrational data.

Free-Free Data

The numerical and experimental results for the in-plane, flee-flee natural frequencies of the I-specimen are presented in Table 9. The maximum error occurs in the first

two modes where the errors are -3.5 and -8.2 % respectively. The discrepancies of the

analysis with respect to the experiment in the first two modes are attributed to the stiffening influence of the elastic bands used to suspend the specimen. The elastic bands act

like compliant springs and in the lower modes the encrgy in the elastic bands represents

a sizable portion of the total energy in the mode. The higher modes have larger frequencies and consequently higher energies, thus, the amount of energy in the elastic

bands relative to the specimen is less significant. The analytical predictions and experimental results for the four, five, and six node modes correlate quite well.

Correlation of Numerical and Experimental Results 135
Table 9. Correlation of the free-free in-plane natural frequencies of the l-specimen

No. of Fxperiment Analysis _ Error b

Nodes tlz Hz %

2 25.8 24.9 -3.5

3 78.3 71.9 -8.2

4 151.0 150.4 -0.4

5 259.0 257.1 -0.7

6 388.0 390.9 0.7

Continuum solution

(Analysis-Exp)/Exp x I00

The predicted four and five noded mode shapes of the l-specimen are shown in Fig.

45. The four and five node modes are presented because they provide an interesting test

of the analytical solution. The experimentally located nodes are indicated by dots in the

figure, the undeformed frame is represented by the solid line, and the analytical mode

shape by the dashed line. The mode shapes were normalized to make the largest displacement 10 % of the radius. Thus, the mode shapes are exaggerated to make them

more visible. The analytical mode shapes correlate quite well with the experimental data.

The mode shapes for the two and three node modes are presented in Appendix A.

Clamped-Clamped Tests

The finite element program was used to model the clamped-clamped vibrational

tests for both specimens. The continuum solution confirmed the predictions of the tinite

element solution for the in-plane response of the I-specimen. Initially, the analyses were

run with clamped-clamped end conditions resulting in predictions which were consistently too high. Modeling the end conditions as hinges with respect to in-plane and outof-plane rotations resulted in predictions which were generally too low. Thus, predictions

using clamped-end and pinned-end conditions bracketed the experimental results indiCorrelation of Numerical and Experimental Results 136
0'

0

/I '_\I

--- t,K_ St'-IAPE

o. • NCOELOCA'TION

(A) FOUR NODE MODE

Figure 45.

!'

\

o

o _

--- M(X_ 8t-t4_

• ** NOOELOCATI_

(B) rive NODE MODE

Free-free in-plane mode shapes for the I-specimen: The predicted mode shapes for the four

and five node modes for the I-specimen are presented with the experimentally located

nodes which are indicated by dots. The undeformed frames are represented by the solid

lines and the analytically predicted mode shape by the dashed lines.

Correlation of Numerical and Experimental Results 137
cating that the experimental boundary conditions acted like torsional springs. In the

lower modes, the experimental results correlated well with the clamped-clamped predictions. In the higher modes, the experimental values correlated well with the pinnedpinned predictions. This indicates that as the frequency and energy of the modes

increased, the clamping effectiveness of the boundary conditions decreased. In general,

the numerical results from the finite element computer program correlated well with the

experimental results with the exception of the in-plane modes of the channel specimen.

The comparison of the in-plane results will be presented first followed by the out-ofplane results.

The predictions for the in-plane natural ti'equencies for the I-specimen from tile finite element program are presented in Fig. 46. The dashed lines correspond to the analytical predictions obtained using clamped-end and pinned-end conditions. The first two

experimental values correlate very well with the clamped-clamped predictions, but the

experimental response begins to shift towards the pinned-pinned predictions in subsequent modes. The last in-plane mode (8 nodes) correlates well with the corresponding

pinned-pinned prediction. In the higher modes, the relative difference between the frequencies for the pinned-end and clamped-end conditions decreases. For the eight noded

in-plane mode, the difference between the frequencies for the clamped-end and pinnedend conditions is just slightly more than 6 %.

The experimental results and analytical predictions from the finite element program

and continuum solutions are presented in Table 10. The predictions from the continuum

solution essentially duplicate the predictions from the finite element code. The classical

solutions from Den Hartog (Ref. 11) for the first in-plane natural frequencies for

clamped and pinned boundary conditions are 59.4 and 29.9 hertz, respectively. These

values correlate very well with the numerical predictions. The correlation between the

Correlation of Numerical and Experimental Results 138
I000

800

6OO

200

,

0

?

/+

//

//

f /'

/"

......... I ......... i ......... i ......... i .........

2 q 6 8

NO.OF NOOL_

-----.EXPERlvE_AL

o-_-., CLAMPED

_-, PI_ED

Figure 46. Clamped-clamped in-plane natural frequencies for the l-specimen: The analytical predictions for the clamped-end and pinned-end conditions are represented by the dashed lines

and the experimental values are the solid line. The modes are ordered by the number of

nodes circumferentially and the frequencies are in hertz.

Correlation of Numerical and Experimental Results 139
analytical predictions for clamped-end conditions and the experimental values is good

initially, but degrades in the higher modes.

Table 10. Correlation of clamped-clamped in-plane natural frequencies for the l-specimen

Analyses

Experiment Clamped Pinned

No. of FEM ' CntnnP FEM _ Cntnm b

Nodes Hz Hz Hz Hz Hz

3 57.7 59.0 58.6 30.6 30,3

4 127.2 129.3 128.7 93.2 92.5

5 233.0 239.9 238.9 187.8 186.6

6 350.6 365.8 364.6 305.6 303.7

7 490.0 529.5 527.7 454.0 451.2

8 617.0 699.2 696.8 622.6 618.3

Finite element model from Ref. 7

Continuum solution

The continuum analysis was easily adapted to model the end conditions as torsional

springs. Modeling the end conditions as torsional springs provided the opportunity to

match the experimental frequency for each mode. The results of this eftbrt are presented

in Table 11. The last column is tile effective torsional spring stiffness necessary to obtain

the analytical natural frequency which essentially matches the experimental value. The

results show that the effective stiffness of the end conditions decreases with increasing

frequency producing the transition from clamped to pinned boundary conditions.

Correlation of Numerical and Experimental Results 140
Table 1I. Effects of boundary spring stiffness on the in-plane vibrational response of the I-specimen

Experiment Continuum Error _

Boundary

Spring

Stiffness

No. of a,

Nodes Hz Hz % in-ib/rad

3 57.5 57.48 -.035 9 x 105

4 127.2 127.32 0.094 9 x 105

5 233.0 233.60 0.258 6 x l0 s

6 350.6 348.58 -.576 3 x 105

7 490.0 485.38 -.951 1 x 10s

8 617.0 618.20 0.200 1 x 103

(Analysis-Exp)/Exp x 100

The curve in Fig. 47 shows tile eft_ct of the torsional spring constant on the prediction of the first in-plane natural frequency. The two horizontal lines represent the

predictions for clamped-cnd and pinned-end conditions from the finite element program.

The logarithm to the base ten of the spring stiffness is plotted on the horizontal axis.

For very compliant and very stiff springs, the predicted natural frequency approaches the

predictions ibr pinned-end and clamped-end conditions, respectively. The plus sign in the

figure indicates the experimental value. The in-plane natural frequencies are sensitive to

small changes in the torsional spring stiffness.

The experimental and numerical results for the in-plane natural frequencies for the

channel specimen are presented in Table 12 and in Fig. 48. The correlation between the

finite element predictions and the experimental results is poor. The predictions for the

natural frequencies using both clamped-end and pinned-end conditions are much lower

than the experimental results. A sensitivity analysis showed that the in-plane natural

frequencies are very sensitive to the value of tile first sectorial moment Io_y.This term

Correlation of Numerical and Experimental Results 141
75

3O

15  ̧

• ! " ! " I " m " a • | • a " e • e • I

0 ! 2 3 q 5 8 7 8 9 10

LOG nN-Le.mAD

___ --- PH4ED

..... _ .,-,.-,.

Figure 47, Effects of torsional spring end conditions: The solid line indicates the effect of the torsional

spring stiffness on the frequency of the first in-plane mode of the l-specimen. The horizontal lines are the predictions obtained using clamped and pinned boundary conditions.

The log of the torsional spring stiffness is plotted on the horizontal axis.

Correlation of Numerical and Experimental Results 142
500'

qO0

300

M.

200

100

o

0

......... | ......... I ......... i ......... I .........

2 q 6 8

NO. OF NODES

---, EXPERI6ENTAL

_,-_ CLAMPED

-_ PI_ED

Figure 48. Clamped-clamped, in-plane natural frequencies for the channel specimen: The analytical

predictions for the clamped-end and pinned-end conditions are represented by the dashed

lines and the experimental values are the solid line. The modes are ordered by the number

of nodes circumferentially and the frequencies are in hertz.

Correlation of Numerical and Experimental Results 143
couples warping and in-plane bending. Other section properties characteristic of an

asymmetric cross section did not have a significant effect on the in-plane response.

Table 12. Correlation of clamped-clamped_ in-plane natural frequencies for the channel specimen

FEM'

Experiment Clamped Pinned

No. of

Nodes Hz Hz Hz

3 44.6 18.8 I0.0

4 99.1 41.5 35.6

5 176.0 76.7 61.9

6 276.7 118.3 107.2

7 413.0 171.3 151.8

Finite element model from Ref. 7

The mode shapes ibr the five and six noded in-plane modes fbr both specimens are

presented schematically in Figs. 49 and 50. The five and six node modes are used for the

comparison because they provide an interesting test of the analysis. The mode shapes

are shown in top and front views. In the top view, the reference axis of the undeformed

frame is a straight line. The l-specimen mode shapes show pure radial motion while the

mode shapes of the channel specimen exhibit both radial and lateral motions. The mode

shapes for the I-specimen correlate very well with the experimental results. The mode

shapes for the channel specimen agree fairly well despite the poor predictions for the

natural frequencies.

The predictions from the finite element computer program for the out-of-plane natural frequencies for the l-specimen are presented with the corresponding experimental

results in Fig. 51. The softening response exhibited by the experimental data seems to

be attributable to the boundary conditions. The experimental data follows the clampedCorrelation of Numerical and Experimental Results 144
_4

0

0

o.* N(I_ELC_ATION ""F_LDCAT_

(A)I-SPECIMEN

8O

Figure 49.

0

0

--._ ....... w-

;.S \

_ --- UIOI_ 81"iAPE

•.. NOI_LO_TION ,"'FOFI_I.O_TION

Five noded, in-plane, clamped-clamped mode shape for both specimens: The five noded

in-plane mode shapes for both sections are shown schematically. The experimentally located nodes are indicated by dots, the undeformed frames are represented by the solid lines

and the analytically predicted mode shape by the dashed lines.

Correlation of Numerical and Experimental Results 145
85,

o_ °-" •

-- _ --- MOOESHAPE

• • • NOOELOCATION •,, FOR_ LOCAI"ICN

(A) I-SPECIM_N

Figure 50.

o

0

i ° _%

0o

o.. NOOELOCAI"K)N ,,,_LOGA'TI_

Six noded, in-plane, clamped-clamped mode shape for both specimens: The six noded inplane mode shapes for both specimens are shown schematically. The experimentally located nodes are indicated by dots, the undeformed frames are represented by the solid lines

and the analytically predicted mode shape by the dashed lines.

Correlation of Numerical and Experimental Results 146
500

qO0

0 3OO

lO0

/

/

/

f

/

/

/

/

/

/

/

,,F

,/

2 q 6 8 tO

NO. OF NOOES

EXPERIVENTAL

_ P_

Figure 51. Clamped-clamped out-of-plane natural frequencies for the l-specimen: The analytical

predictions for the clamped-end and pinned-end conditions are represented by the dashed

lines and the experimental values are the solid line. The modes are ordered by the number

of nodes circumferentially and the frequencies are in hertz.

Correlation of Numerical and Experimental Results 147
end predictions through the first five modes, then shifts from clamped-end to pinned-end

conditions between the fifth and sixth modes. This transition is very abrupt compared

to the shift observed in the in-plane response. After the shift, the experimental data follows the pinned-end predictions closely.

A plot of the analytical predictions and the experimental results for the out-of-plane

natural fi*equencies of the channel specimen is presented in Fig. 52, Due to the asymmetric cross section of the channel specimen, predictions for only the first six out-ofplane natural frequencies were obtained. The transition from clamped-end to pinned-end

conditions was also observed in the out-of-plane response of the channel specimen. The

finite element predictions agree well with the experimental data through the first three

modes. The transition from effective clamped-end to effective pinned-end conditions

begins in the fourth and fifth modes and is completed between the fifth and sixth modes.

It appears that the pinned-end predictions would correlate with the higher experimental

natural frequencies if those predictions were available.

The numerical and experimental results for the out-of-plane natural fi'equencies for

both specimens are summarized in Table 13. Though the plots show good correlation

in the first few modes, the discrepancy of the predicted frequency with respect to the

experimental value for the fundamental mode is actually 8.5 % and 6.7 % for the Ispecimen and channel specimen, respectively. The classical solution from Brown (Ref.

12) predicts a frequency of 7.5 Hz for the fundamental frequency of the I-specimen

confirming the prediction from the finite element code. The prediction of 15.4 I-tz obtained from the out-of-plane continuum solution is almost 100 % too high. The fact that

none of the analyses can predict the fundamental frequency is disturbing. Generally,

experimental frequencies are lower than those from the analysis. In this case, the experCorrelation of Numerical and Experimental Results 148
500

qO0

0 3oo

200,

0

//

/J /

2 q 6 8 I0 12

NO. OF NODES

•---- EXPER_F.I_AL

_-_ Ct._

_ PI_ED

Figure 52. Clamped-clamped out-of-plane natural frequencies for the channel specimen: The analytical predictions for the clamped-end and pinned-end conditions are represented by the

dashed lines and the experimental values are the solid line. The modes are ordered by the

number of nodes circunfferentially and the frequencies are in hertz.

Correlation of Numerical and Experimental Results 149
Table 13. Correlation of clamped-clamped, out-of-plane natural frequencies for both specimens

FEM a

Experiment Clamped Pinned

No. of

Specimen Nodes Hz Hz Hz

I-specimen

Channel

2 8.2 7.5

3 29.8 25.4 6.3

4 64.3 58.4 24.6

5 110.4 105.4 57.7

6 157.8 161.7 104.1

7 164.7 220.7 158.5

8 207.4 278.2 214.9

9 254.7 332.0 269.8

10 314.7 383.8 322.0

2 7.4 6.9

3 25.4 24.2

4 55.4 55.8

5 91.1 103.7

6 148.8 164.4

7 155.2 234.8

8 187.0

9 214.0

5.9

23.1

54.7

100.4

156.6

Finite element model from Ref. 7

Correlation of Numerical and Experimental Results 150
imentalresultsarehigherthan the analysis,which is not usually the casewhen correlating analysisandexperiment.Thisdiscrepancyremainsunresolved.

]'he modeshapesfor the five and six nodeout-of-planemodesfor both specimens

arepresentedin Figs.53and 54, respectively. The mode shapes are shown in both top

and front views. The mode shapes for the channel specimen modes show radial displacements as well as out-of-plane displacements. This is typical of the coupled response

of the channel specimen. The mode shapes for the l-specimen show pure out-of-plane

motion characteristic of an uncoupled response. Good correlation is observed between

the predicted mode shapes and the experimentally located nodes for the mode shapes

of both specimens. The correlation for the lower modes was generally as good or better.

The mode shapes for the other modes are presented in Appendix A.

Correlation of Static Data

To correlate the experimental flexibility coefficients with the analytical values it is

necessary to choose the experimental results which are most representative of the small

deflection response of the frames. The in-plane test results are considered more reliable

than the out-of-plane test results, therefore, the values for flexibility coefficients

_,a, _2a, and _33 are taken from the in-plane tests. Specifically, the experimental values

associated with the forty pound radial load are used in the correlation of the data for

both specimens. The coefficients from the forty pound radial load were chosen because

it was the lowest load with the least amount of scatter in the data associated with it. In

the cases where the values for positive and negative torques are close, the values are

averaged. When the magnitudes are significantly different, the value associated with the

Correlation of Numerical and Experimental Results 151
0

0

• • • NGDE LOCA'I"ICN • • ' FCF_E LOCATION

(A) I- SPECIM._

Figure 53.

N

O'

00

m .--.

_ --- t,OC_ 8HAPE

ooo NOI_LOGATION '"FOR_LOCATION

Five noded, out-of-plane, clamped-clamped mode shape for both specimens: The five noded

out-of-plane mode shapes of both specimens are presented schematically. The experimentally located nodes are indicated by dots, the undeformed frames are represented by

the solid lines and the analytically predicted mode shape by the dashed lines.

Correlation of Numerical and Experimental Results 152
e$,

0

0

A •

_ --- MOOESHAPE

ooo NOI_LOCATION '"FC_CELOCATION

(A)I-SPEC_M_

Figure 54.

0'

0

s _.

! v v

v v _

*** NOOELOCATION "" FOR_ LOGATION

{B}_ sPzc'm_

Six noded, out-of-plane, clamped-clamped mode shape for both specimens: The six noded

out-of-plane mode shapes of both specimens are shown schematically. The e×perimenta]ly

located nodes are indicated by dots, the undeformed frames are represented by the solid

lines and the analytically predicted mode shapc by the dashed lines.

Correlation of Numerical and Experimental Results 153
negative torque is used in the correlation. The values for flexibility coefficients

c%, _22, and _32 are only obtained from the out-of-plane tests. The values for these coefficients associated with lateral loads of 3.25 and 1.063 pounds are used for the Ispecimen and channel specimen, respectively.

The expcrimental values for the flexibility coefficients are correlated primarily with

predictions from the finite element solution. The predicted values were obtained using

clamped boundary conditions and two types of section properties. The finite element

program was developed for isotropic, homogeneous, materials. Purely geometric section

properties are generally used as input to describe the cross section of the specimen. To

try to account for the composite material system the modulus-weighted section properties discussed in Chapter 5 are also used. Thus, predictions obtained using modulusweighted and geometric section properties are presented. The modulus-weighted section

properties and the geometric section properties for both specimens are presented in Appendix C.

The finite element computer program uses linear elastic structural theory, thus, the

predicted flexibility matrix is symmetric. The experimental tests measured the total response of the frames which is a combination of linear and nonlinear responses. As a result, the experimental tlexibility matrices are generally not symmetric. The flexibility

influence coefficients of primary interest are the diagonal terms; _H, _2, and c%. These

terms play the largest role in describing the load carrying behavior of the frames. With

the exception of the c_23term, the off-diagonal terms play a small role in the static response of the frames. The signiticance of _23 is a result of the coupling between out-ofplane bending and torsion due to the curved geometry. The correlation of the static

results for the I-specimen will be discussed first followed by the results of the channel

specimen.

Correlation of Numerical and Experimental Results 154
1-Specimen Results

The flexibility coetlicient 0_11fog the I-specimen is a special case in that values for

(_11were obtained from two independent tests and from two independent analyses. The

analytical predictions and experimental results for the flexibility coeft]cient cq_ are presented in Table 14. The finite element predictions using clampcd and pinned boundary

conditions bracket the experimental value from the in-plane static test, and also the

crush test. The crush test result for _n is closer to the prediction obtained for clampedend conditions reflecting the difference in the experimental boundary conditions between

the tests. The continuum static solution was used to model the end conditions as

torsional springs. The torsional spring stiffness required to match the in-plane static test

value for c% is smaller than the torsional spring stifliaesses required to match the in-plane

natural frequencies in Table 11. The lower torsional spring stiffness required in the static

tests with respect to the vibrational test is probably due to the difference in the static

mode shape and the vibrational mode shape.

Table 14. Correlation of flexibility coefficient _,,for the l-specimen

_11

Source in/lb

Experiment

Crush Test 0.640 x 10-3

Static Test 0.843 x 10-3

FEM _

Pinned 0.926 x 10-3

Clamped 0.578 x 10-3

Continuum

Spring b 0.8559 x 10 -3

Finite element solution from Ref. 7

Continuum solution with torsional spring stiffness

_ = 2 x 104 in-lb/rad.

Correlation of Numerical and Experimental Results 155
The effectof the torsional springconstanton the valueof flexibility coefficienti_11

is demonstratedin Fig. 55. The logarithm to tile base ten of the spring stiffness is plotted

along the horizontal axis and _. is plotted on the vertical axis. The two horizontal lines

represent the predictions from the finite element program for clamped-end and pinnedend conditions. For very compliant springs and very stiff springs, the continuum analysis matches the predictions of the finite element code for pinned-end and clamped-end

conditions, respectively. The plus sign in the figure indicates the experimental result, and

it is closer to the pinned-end prediction than to the clamped-end prediction. Comparing

the plots in Figs. 47 and 55 shows that the torsional spring stiffness required to achieve

effectively clamped-end conditions are different for static and dynamic conditions. The

effective stiffness of the experimental boundary conditions is higher for the dynamic tests

than for the static tests.

The flexibility coefficients for the I-specimen from the finite element program using

both modulus-weighted and geometric section properties are presented with the corresponding experimental values in Table 15. In general, the correlation between the experimental results and the predictions obtained using the modulus-weighted section

properties is better than with the predictions obtained using the geometric section

properties. The modulus-weighted values predict a stiffer response than the geometric

section properties.

The discrepancy in the flexibility coefficient 0Cllis attributed to the flexibility in the

experimental boundary conditions. The numerical predictions for flexibility coefficient

_22 are significantly higher than the experimental result. The modulus-weighted value

correlates the best, but it is 20 % higher (more compliant) than the experimental value.

Torsional springs cannot explain this discrepancy since additional flexibility would make

the predictions more compliant. This result is consistent with the discrepancy in the

Correlation of Numerical and Experimental Results 156
.0012

.0009

,0006,

.0003

0 .....1 2 3, .4. 5, . 6, ._;. _. 9, .

LOG H.B/RAD

...... _ _ BO_B_Bcr

i

10

Figure 55. Influence of torsional spring stiffness on ohz: The plot shows that the value ofall is very

sensitive to small changes in the flexibility of the end conditions. The log of the torsional

spring stiffness is plotted on the horizontal axis. The horizontal lines are the predictions for

clamped-end and pinned-end conditions.

Correlation of Numerical and Experimental Results 157
fundamental out-of-plane natural frequency of the I-specimen. The predicted values for

flexibility coefficients _a3 and _23 using modulus-weighted section properties correlate

well with the experimental values. The predictions obtained using geometric section

properties do not correlate as well. The correlation with the experimental result for _32

from the out-of-plane tests is not as good. The discrepancy is on the order of 20 %

similar to the discrepancy in _22.

Table 15. Correlation of the flexibility coefficients for the l-specimen

Flexibility Experiment Analysis"

Coefficient M-W b Geometric c

C(.ll in/lb 0.843 x 10 -3 0.5778 x 10 -3 0.6709 x 10-3

c_22 in/lb 0.113 0.1406 0.2006

¢33 rad/in-lb 0.131 x 10-2 0.1396 x 10 -2 0.1892 x 10 -2

_23 l/Ib 0.101 x I0 -l 0.1109 x 10 -I 0.1503 x 10 -1

cq2 I/Ib 0.0877 x 10 -1 0.1109 x 10 -1 0.1503 x 10-1

_2 in/lb 0.148 x 10-2 0.715 x 10-s 0.834 x 10 -s

c_2_ in/lb 0.658 x 10-a 0.715 x 10 -s 0.834 x 10-5

_3 l/lb 0.107 x 10 -3 0.197 x lO-6 0.497 x 10-6

_31 l/lb 0.261 x 10-4 0.197 x 10-6 0.497 x 10-6

Finite element solution from Ref. 7

Modulus weighted section properties

Geometric section properties

The predicted values for flexibility coefficients cq2 and _3 are very small and can be

interpreted as zero. The experimcntal value for flexibility coefficient cq2 does not correlate well. This discrepancy is believed to be due to a geometric nonlinearity which cannot

be modeled using linear structural theory. The experimental values for c_2_,oq3, and Cql

are small and were inconsistently measured. The scatter in the data associated with these

coefficients indicates that the effects of these coefficients are too small to measure reliably. Thus, the discrepancies in these coefficients is not significant.

Correlation of Numerical and Experimental Results 158
Channel Specimen Results

The flexibility coefficients for tile channel specimen from the finite element program

using both modulus-weighted and geometric section properties are presented with the

corresponding experimental values in Table 16. In general, the correlation between the

experimental results and the predictions obtained using the modulus-weighted section

properties is better than with tile predictions obtaincd using the geometric section

properties. The predicted values for flexibility coett_cient cql are a minimum of five times

larger than the expcrimental result. This discrepancy is too large to be explaincd by

experimental anomalies. The discrepancy is consistent with the correlation observed in

the in-plane natural frequencies of the channel specimen. A sensitivity analysis revcaled

that _1_ is very sensitive to the first sectorial momcnt I,oy. Other section properties

characteristic of an asymmetric section did not significantly effect the prediction of cql.

The discrepancy in the values for flexibility coefficient _22 is consistent with the discrepancy in the fundamental out-of-plane natural frequency of the channel specimcn.

The discrepancy is smaller than the corresponding discrepancy in the l-specimen results.

The experimental values for flexibility coetticicnts cc33and c_23corresponding to tile negative torque data correlate well with the modulus-weighted predictions. The experimental value for c_32obtained from the out-of-plane tests does not corrclate well with

either of the predictcd values.

Correlation of Numerical and Experimental Results 159
Table 16. Correlation of tile flexibility coefficients for the channel specimen

Flexibility Experiment

Coefficient

Analysis '

'V b Geometric c

C_ll in/lb 1.66 X 10-3

0(22 in/lb 0.242

_33 rad/in-lb 0.258 x 10-2

_23 l/lb 0.224 x 10-1

_32 l/lb 0.120 x 10-1

cz._2 in/lb 0.842 x 10-2

_22 in/lb 0.790 x 10-4

_3 1/[b 0.369 x 10 -2

c_31 1/lb -.201 x 10 -4

8.013 x 10 -3

0.2678

0.2475 x 10 -2

0.197 x 10-_

0.197 x 10-2

1.456 x 10 -2

1.456 x 10 -2

0.407 x 10 4

0.407 x 10 -4

9.758 x 10-1

0.3653

0.3401 x 10..2

0.271 x 10-2

0.271 x 10-_

1.777 x 10-2

1.777 x 10 -2

0.497 x 10 _

0.497 x 10 -4

Finite element solution from ReE 7

Modulus weighted section properties

Geometric section properties

The correlation of flexibility coefficients 1_-12and _31 is not bad. The values are at

least on the same order of magnitude. However, the good correlation is considered a

coincidence since the experimental value for e_2 is likely measuring a geometrically nonlinear response and the scatter in the data associated with u3_ is very high. Even the

predicted values for cq2 and _3_ should be considered carefully since they are linked to the

poorly predicted in-plane response of the channel specimen. The experimental values for

flexibility coefficients ch_ and cq3 do not correlate well with the predicted values. The experimental values are actually smaller than the corrcsponding values for the I-specimen.

This is unusual since these coefficients couple the in-plane and out-of-plane responses

and the coupling for the channel specimen is expected to be more pronounced than for

the l-specimen. This indicates that the experimental values for these coefficients cannot

be relied upon. Certainly the correlation in these coefficients cannot be treated seriously.

Correlation of Numerical and Experimental Results 160
Chapter 7

Concluding Remarks

The primary purpose in this study was to evaluate a new finite element program

which was developed by Noor et al., (Ref. 7) specifically for the free-vibrational analysis

of curved, thin-walled beams with open sections. An experimental program was undertaken to generate data which was characteristic of the static and free-vibrational response of two test specimens. The test specimens were semi-circular, graphite-epoxy

frames which were intended to represent aircraft fuselage frames. One test specimen had

a symmetric I cross section and the other had an asymmetric channel cross section.

The experimental program consisted of both static and dynamic tests. A series of

static tests generated the flexibility matrix which relates radial, lateral, and torsional

loads at the midspan to radial, lateral, and twist displacements at the midspan. The dynamic tests provided the frequencies and node locations of the in-plane and out-of-plane

free-vibrational modes for clamped-clamped end conditions.

The finite element program is based on a Vlasov-type, thin-walled, curved beam

theory and accounts for the additional effects of transverse shear deformation and

Concluding Remarks 161
rotatory inertia. Ilowever, for the specimensin this study, transverseshearingdeformationsdid not significantlycontributeto the response.The analyticalpredictionsfrom

the finite elementprogramwereconfirmedin part by independentcontinuumsolutions

for the in-planestaticand dynamicresponseof a monosymmetriccurvedbeam.

Theanalyticalpredictionscorrelatedwell with the experimentaldatafrom the symmetric I-specimen. It was tbund that the experimentalboundary conditionsactedlike

torsional springsinsteadof clamps.Theanalyticalpredictionsfor the naturalfrequencies

usingclamped-endand pinned-endconditionsbracketedthe experimentaldata.Theanalytically predictedmode shapescorrelatedquite well with the experimentallylocated

node positions.The largestdiscrepancyoccurredin the prediction of the first out-ofplanenatural frequency,which was8.5% lowerthan the experimentalvalue(seeTable

13).

The analyticalpredictionsfor the static responseof the I-specimenalso correlated

well with experimentaldata.A substantialdiscrepancyin flexibility coefficientc% (radial

displacement/radial load) was shown to be a tunction of the flexibility in the experimental bounda_ conditions. Further investigation showed that the static tests required

a softer support to model the boundary conditions than was required in the dynamic

tests (compare Tables 1 1 and 14). The analytical prediction for flexibility coefficient "22

(lateral displacement/lateral load) was larger than the experimental value. Yhis is consistent with the discrepancy in the first out-of-plane natural frequency, since in both

cases the analysis predicts a more compliant response than is seen in the experiments.

The correlation between analysis and experiment is less satisfactory for the channel

specimen than for the I-specimen. In general, the predicted out-of-plane static and

vibrational responses correlated well with the experimental data. The predictions for the

out-of-plane natural frequencies using clamped-end and pinned-end conditions bracketed the experimental results (see Fig. 53). The predictions for flexibility coefficients

Concluding Remarks 162
a33and _23correlatedwell with the experimentaldata( seeTable 16). The analytical

predictionsfor the first out-of-planenatural frequency(Table 13)and flexibility coefficient c% (Table 16) were again predicted to be more compliant than the experimental

results consistent with the 1-specimen results. These discrepancies are worth noting since

it is unusual when correlating analysis with experimental data to find the analysis more

compliant than the experimental response.

Large discrepancies were observed between the analytical predictions for the inplane static and vibrational responses of the channel specimen and the experimental

data. The experimental natural frequencies are two to three times higher than the predicted values (see Table 12). Thus, the analysis is signiticantly more compliant than the

experiment. Consistent with this, the analytical prediction for flexibility coefficient _,_

(Table 16) is five times higher than the experimental values. Discrepancies of this magnitude are difficult to explain, however, further investigation did show that the in-plane

response of the asymmetric channel specimen is extremely sensitive to the cross-sectional

property I_y which couples in-plane bending and warping.

The finite element program was developed for isotropic homogeneous materials.

The composite nature of the test specimens was accounted for in part by using

modulus-weighted section properties in place of section properties calculated based

purely on the geometry of the cross section. The flexibility influence coefficients obtained

using the modulus-weighted section properties correlated better with the experimental

data than did coefficients obtained using purely geometric section properties (see Tables

15 and 16).

The magnitudes of some of the experimentally measured flexibility coefficients were

found to be a function of the algebraic sign of the applied torque (see Table 6 for example). The experimental values for flexibility coefficients _,2 and _, differed by two

orders of magnitude (Table 5-8). The values for flexibility coefficients cq3 and e% were

ConcludingRemarks 163
differentmagnitudesand in the caseof the channelspecimenwereconsistentlyof differentsigns.Linear structuraltheorypredictsthe flexibility matrix to besymmetric,and

consequentlycannotaccountfor the asynunetryof the flexibility matrixesmeasuredin

theexperiments.A detailedanalysisof the static out-of-planeresponseof curvedfi'ames

requiresa solutioncapableof modelinggeometricnonlinearities.Thisis particularlytrue

for frameswith asymmetriccrosssections.Thenecessityfor a nonlinearsolutionis less

apparentin the vibrational responseof the curvedframes.

Future Work

The data from the experimental tests could be improved upon in a number of ways.

Simply securing the ends in potting compound and clamping the ends to a steel beam

was not sufticient to model clamped end conditions. More elaborate steps could be taken

to obtain clamped conditions. As a minimum, the potted ends should be bolted to the

steel beam. The resolution in the out-of-plane static tests might be increased if the

specimens were loaded with less force and more torque. This requires using larger moment arms. Doing this without adding excessive weight to the specimens might not be

a trivial exercise.

With respect to the analysis, this study has raised some serious doubt about the

accuracy of the predictions from the finite element program for the in-plane response

of a curved beam with an asymmetric cross section. Comparisons with similar experimental data for other asymmetric sections seems to be in order. Certainly, some effort

should be made toward resolving this issue since this element is expected to be implemented in a larger finite element code.

Concluding Remarks 164
The discrepanciesin the in-planeresponseof the channelspecimenareof a magnitudethat makesit diflicult to attribute themto thecompositematerialsystem;however,

extendingthe ttooke's law relation in the finite elementprogram to account for the

constitutiverelationsfor laminatedcompositesmight help.Doing this may help explain

the discrepancies in the fundamental out-of-plane natural frequencies, since they are of

a much smaller magnitude.

Thin-walled curved beam theories are complicated and application to composite

structures should be undertaken with some caution. The assumption in Vlasov's theory

that the shear strain in the midplane of the wall vanishes is a result ofisotropic elasticity.

This assumption should be re-examined using anisotropic elasticity to determine if it is

a valid assumption with respect to laminated composites.

Finally, a logical extension of this experimental work would be to load the frames

to failure. Such tests would provide information on the large displacement response of

the composite frames as well as first failure and post failure response. Research of this

nature is certainly more germane to the issue of crashworthiness. This research is currently being conducted by Mr. E. Moas, Graduate Research Assistant, and Professor

O. It. Griffin in cooperation with the Landing and Impact Dynamics Branch at NASA

Langley Research Center through the NASA-Virginia Tech Composites Program.

Concluding Remarks 165
REFERENCES

°

.

°

4_

.

.

o

.

°

10.

11.

Farley, G. L., "Energy Absorption of Composite Materials," Journal of Composite

31aterials, Vol. 17, 1983.

Williams, M. S. and tlayduk, R. J., "Vertical Drop Test of a Transport Fuselage

Section Located Forward of the Wing," NASA Technical Memorandum 85679, August, 1983.

Williams, M. S. and Hayduk, R. J., "Vertical Drop Test of a Transport Fuselage

Center Section Including the Wheel Wells," NASA Technical lllemorandum 85706,

October, 1983.

Fasanella, E. L. and Alfaro-Bou, E., "Vertical Drop Test of a Transport Fuselage

Section Located Aft of the Wing," NASA Technical Memorandum 89025, September,

1986.

Boitnott, R. L. and Carden, H. D., "Drop Testing and Analysis of Six-Foot Diameter Graphite-Epoxy Frames," in tire Proceedings of tlre AHS National Specialists

llIeeting on Crashworthy Design of Rotorcraft, April, 1986.

Boitnott, R. L. and Fasanella, E. L., "Impact Evaluation of Composite Floor

Sections," SAE Technical Paper Series 891018, April, 1989.

Noor, A. K., Peters, J. M., and Min, Byung-Jin, "Mixed Finite-Element Models for Free

Vibrations of Thin-Walled Beams," NASA Technical Paper 2868, February 1989.

Hoppe, R., "The Bending Vibration of a Circular Ring," Crelle Jom'nal of Mathematics, Vol. 73, 1871, pp. 158.

Lamb, "On the Flexure and tile Vibration of a Curved Bar," Proceedings of the

London Mathematical Society, Vol. 19, 1888, pp. 365-376.

Love, A., A Treatise on tile Mathematical Theory of Elasticity, Fourth Edition,

Dover Publications, New York, 1944, pp. 451-454.

Den Hartog, J. P., "The Lowest Natural Frequcncy of Circular Arcs," Philosophical

ll,Iagazine, Ser. 7, Vol. 5, 1928, pp. 400-408.

REFERENCES 166
12. Brown, F. I|., "Lateral Vibration of Ring-Shaped Frames," Journal of the Franklin

Institute, Vol. 218, July 1934, pp. 41-48.

13. Timoshenko, S., "Theory of Bending, Torsion, and Buckling of Thin-Walled Members of Open Cross Section," Journal of the Franklin Institute, Vol. 236, No. 3, pp

201-219; No. 4, pp. 249-268; No.5, pp. 343-361, 1945.

14. Lang, T. E., "Vibrations of Thin Circular Rings, Part I. Solutions for Modal

Characteristics and Forced Excitation," Technical Report No. 32-261, (contract no.

NAS7°100) Jet Propulsion Laboratory, California Institute of Technology, July

1962.

15. Lang, T. E., "Vibrations of Thin Circular Rings, Part II, Modal Functions and

Eigenvalues of Constrained Semi-Circular Rings," Technical Report No. 32-261,

(contract no. NAST-100) Jet Propulsion Laboratory, California Institute of Technology, March 1963.

16. Maddox, N. R., "Free Oscillations of Thin-Walled Open Section Circular Rings,"

Doctorate of Philosophy Dissertation, Georgia Institute of Technology, Atlanta,

Georgia, October 1969.

17. Hammoud, A. S. and Archer, R. R., "On the Free Vibrations of Complete and Incomplete Rings," Developments in Mechanics, Vol. 2, Part 2, April 1963, pp. 489-524.

18. Endo, M., "Flexural Vibrations of a Ring with Arbitrary Cross Section," Bulletin of

the Japan Society of Mechanical Engineers, Vol. 15, 1972, pp. 445-454.

19. Kirkhope, J., Bell, R. and Olmstead, J. L. D., "The Vibration of Rings of Unsymmetrical Cross Section," Journal of Sound and Vibration, Vol. 96, 1984, pp. 496-504.

20. Williams, H. E., "On the Equations of Motion of Thin Rings," Journal of Sound and

Vibration, Vol. 26, No.4,1973, pp. 465-488.

21. Rao, S. S., "Effects of Transverse Shear and Rotatory Inertia on the Coupled

Twist-Bending Vibrations of Circular Rings," Journal of Sound and Vibration, Vol.

16, No. 3, June 1971, pp.551-566.

22. Culver, G. C., "Natural Frequencies of Horizontally Curved Beams," Journal of the

Structural Division, ASCE, April 1967, pp. 189-203.

23. Vlasov, V. Z., Thin-Walled Elastic Beams, Second Edition, Moscow, 1959; translated

from Russian and published for the National Science Foundation, Washington,

D.C., 1961.

24. Gardner, T. G. and Bert, C. W., "Vibration of Shear Deformable Rings: Theory and

Experiment," Journal of Sound and Vibration, Vol. 103, 1985, pp. 549-565.

25. Bhimaraddi, A., "Generalized Analysis of Shear Deformable Rings and Curved

Beams," International Journal of Solids Structures, Vol. 24, No. 4, 1988, pp. 363-373.

26. Gjelsvik, A., The Theory of Thin-Walled Beams, Wiley, New York, 1981.

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27.

28.

29.

30.

31.

32.

33.

Bauld,N. R. and Tzeng,L-S.,"A VlasovTheory for Fiber-ReinforcedBeamswith

Thin-Walled OpenCrossSections,"International Journal of Solids Structures, Vol.

20, No. 3, 1984 pp. 277-297.

Lo, P. K., "Comparison of Theory and Experiment for Flcxural-Torsional Buckling

of" Laminated Composite Columns," blaster of Science Thesis, Virgbffa Polytechnic

Institute and State University, Blacksburg, Virginia, 24061, December 1985.

Mabson, G. E., "Effect of Curvature on Composite Material Beams," bt the proceedings of the AIAA/ASME/ASCE/AHSIASC 30th Structures, Structm'al Dynamics

and alaterials Conference, Part 4,1989, Mobile, Alabama, pp. 1779-1787.

Tralli, A., "A Simple Hybrid Model for Torsion and Flexure of Thin-Walled

Beams," Computers & Sn'uctures, Vol. 22, No. 4, 1986, pp. 649-658.

Pindera, M-J., Choksi, G., Hidde, J. S., tlerakovich, C. T., "A Methodology for

Accurate Shear Characterization of Unidirectional Composites," Journal of Composite hlaterials, Vol. 21, 1987, pp. 1164-1184.

Kaplan, W., Ordinary Differential Equations, Addison-Wesley Publishing Co. Inc.,

London, 1958, pp. 231-237.

Coyette, J. P., "An Improved Subroutine for the Estimation of Torsional Properties

of Thin Walled Open Cross Sections," Engineering Computations, Vol. 4, September

1987, pp. 240-242.

REFI"REN CES 168
Appendix A

Vibrational Test Data

This appendix contains some of ttle data from the vibrational tests that was not

presented ill the text. The first section contains the frequency response plots from the

clamped-clamped tests. The second section contains the comparison of the analytical

mode shapes and the experimental node locations.

Frequency Response Data from Dynamic Tests

The frequency response plots from tile structural analyzer dynamic tests are presented here for the interested reader. Two sets of tests were run for each specimen, one

series with radial excitation and one series with lateral excitation. The specimens were

excited over a frequency range of 0-400 hertz. Noise began to dolninate the response

signal at about 37.5 hertz. In order to obtain sufficient resolution, it was necessary to test

the specimens over smaller sub ranges, thus, the plots are presented in stages which

overlap.

Three displacement probes were monitored in each test, thus, there are three plots

in each figure. Each plot consists of a response curve and a phase angle curve. A natural

frequency is marked by a spike in the response curve accompanied by a shift in the phase

angle curve.

The probes were mounted in different orientations to detect both radial and lateral

motion. The probe location and orientation is labeled in each plot. The three different

orientations are defined in Fig. 56. The locations of the probes are measured in inches

from the midspan of the specimen. The horizontal axis is the frequency of excitation in

hertz and the vertical axis for the response curves has units of inches per pound on a log

scale.

TILe I-specimen plots are presented first followed by the channel specimen plots.

Ideally a mode is indicated by a sharp isolated spike, however, frequently the spikes are

small and closely spaced making interpretation difficult. This is particularly true for the

channel specimen plots where both in-plane and out-of-plane modes were excited with

either radial or lateral excitation. The magnitude of the spikes is generally a poor indicator of" the relative strengths of the modes. The spike magnitude is a strong function

of probe location and the location of the electro-magnetic shaker. If the shaker or tile

probe are located near a node, the mode will be difficult to excite or detect, respectively.

The plots are presented without further comment.

Vibrational Test Data 169
L

(A)

! !

(B)

I !

(c)

Figure 56. Proximity probe orientations for clamped-clamped vibrational tests: The probe orientations

relative to both cross sections are presented in (A)-(C). (A) In-plane edge. (B) In-plane

Web. (C) Out-of-plane.

Vibrational Test Data 170
0

0

13

1. 000r_ .... _6.5" OFF CNTR-.:

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Frequency (Hz)

.._ Figure 57. I-specimen frequency response plot for radial excitation; I1-111 I Iz
0

0

E

8

°,.-_

77o 70_

417. Ou

37.89_

I, 000m

340,9u

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/\ 13" OFF CNTR -

_- / ',,,. IN PLANE

rlllll IIIIIIIIII IIIIIIIIIIIIIIIIIII IIIlllIIIlllllllIl IIIIIII III

109,8 _130.0 140z0 __.__150,0 160,9 "--"_.__ _180,0 _^

_,, ,_,,_,, 7,,,, ,,,,,, X,,_,,,,,, _-,, _,_ _-5,_,_}

i A I i i 6.5 OFF cMTR

-_ v V_v -, -y-V_,.D/_A_

_ I I I I I I I I I I I I I I I 1 1 I I I I I I I I I I I I I I I 1 I I I I1.1 I I I I I I I I I I I I I I l I I I I I I I I I I I I 1

109,8 130.0 140,0 150,0 160,0 180,0

Frequency (Hz)

._ Figure 58. I-specimen frequency response plot for radial excitation; 110-180 llz
0

0

U-,

o

13..,

° ,.,=,G

C_

I i I I I I I I .j

_- 0" OFF CMTR _ _.

2. 108m_,. IN PLAME _kf/ _ ......

827. In- i i _ i t i l i l i i i i i i i

233.7 .... _ _ , 300.0 ..... 400. i .-.

_,,,_,_._,--_----.-_

l i i -I i i I l i i i i i , i i

6.570_-_ 13" OFF CMTR

[ _'.-..._^ . IM PLAME ._ ]

I I I I I I 1 I I

1233.7 "-." ,,,.&_ .'L ,2U\l I /v,.300.0--"',,JA?,44 .,4,'v#',- ,_Y-.-",--_.._._ . #x J1£,_ ,400.1 v¢,,,,,_

, , I ' 5" OFF CHTR ' ' , l , , _ , , [

83.15u OUT o_ PLAI'E

i, 376,., t i i i I. , I J i J t _ i i _ i i

233.7 300.0 400.

Frequency (Hz)

.% Figure 59. I-specimen frequency response plot for radial excitation; 233-400 IIz
0

..-..-I

CP_

.,.=4

i. 785_

100.0_

41.13u

1.611

1.802

47.73_

i(llllllllll I I It II II II II III I i I II I|I I IIII I IIII III llllllil I I III II II I I IIII It I

j _ ' ' ' 11" OFF CMTR ' I' :.

!/ _ OUT o' PLANE

?-

_ltl n lin iii1 Ill Inllollinlllilll Illllnllmllltllanllnli.tnlll|n'Lllnnlntl|ln|lliil

6. 835 9,000 10.00 11,00 12,00 13.00 15.03

'(l i'i 'v,, ,_-_-ii ii,,...____. i iii i u_, u i I i,,, i"'_u, ,_111'11 _ll_''U1_l'lU''U1"l' ,liulli, lllV,,_li,.

/" _ 0" OFF CMTR !

- _ _ IN PLAHE !

6.835_-----_9.000 10.00 11.00 m

12.00---13.00 15.03----

Ulll UillilUllllUlllilllllUllili]ll llUllUi l lull I lUlUlUlii"llllll i illlil,,[lll,lll,l ./--______ ,

_- / _ 6.5" OFF CMTR -:

- _._ _-_ IN PLANE --

- EDGE

il_L _lni_f_lon_nnul,nll_nnnn_n_lno ninnninllu,nninnal,n

6. 835 9. 000 10.00 11.00 12.00 13.00 15,03

Frequency (Hz)

._ Figure 60. I-specimen frequency response plot for lateral excitation; 6.8-15 iiz

.la
q.)

O

o

I.L

E

.,,-w

c_

I I I I I 1 I I I I I I I I I I I I I I I I I

' A 11-off CNTR

1. 000rr

100. Ou

B•446u z , , , , i i I i i i i i i i , i I I I i I I

19• 82 100.0 200.0 259.7

J_ "I - ----V ',,,...

I I ' I I I l I J I I I I I I I I l J I I I I I

188.0mE A .i. 0" OFF CHTR

__/ _ , , A, J_ IN PLANE

t_l _ -"-,h-,f_ _.__. I "L . _,_ EDGE /_

99.05u i i i i _ _ I I J i i i J l _ i _ I I I I I I

"19• B2 .......... 100.0 .......... _--_-" "200.0 ''_-259 •7__

• _,.A ._A . ,_ ,

' ' ' ' ' ' ' ' _ ' ' -'- ' '61s,,ok-_-CNi-R-" ' "

__

_ I_]_-_ ]\ EDGE ,, "_

41 ::>4ur

• _ , i I i z I I I , I I I _ _ _ I I I I I I

19.82 100• 0 200.0 259.7

Frequency (Hz)

Figure 61. I-specimen frequency response plot for lateral excitation; 20-260 llz
O

I-,i

O

(,.)

,,--,,I

43.00m

1.000m

30.48u

-f i I I . I I I I I I I

/_, ' 5.5" OFF CNTR ._

r

I I I I I I I I I I I I

i i i i _ i I i i

A A 0" orr C.TR 3

J_^ }\ A_'.__ _/_/ I_ IN PLANE

I I I 1 I I I 1 I I I I

140.6rr

I. 000_

32.52u_, I

30.07

-30. oT-Lr--__/--_ --_ lOO._ -159.

' ' ' ' ' ' I '9" OFF CNTR ' '

I I I I I I I I 1 I I ,,

100.0 159.9

Frequency (Hz)

..% Figure 62. Channel specimen frequency response plot fi)r radial excitation; 30-160 I lz

o_
I=.l

0

0

I::l

3. 250.

129.5u

4. 165_

IIIIIIIIIl'tlllll III111_11111|11111 IIT/IIIIIIIII I111 11111 IIIIII111 I IIIIIIll lllllllllilllllllll

_" 5.5" OFF CNTR ' /_ .....

IIIIIIIIII111111111111111111111111 IIII1111111111111111111111111111111 IIIIIIIIIIIIIIIIIIIIIIIIIII

142. i- -160. B.--.__180.0 ----.200._b_j_____ 220.0 240.0

... _ _./'___

/lllllllllllillf'llllllllllll,llll, Ill II1 II111]'1 IIIII IIIII111111111111 IIIIAIIIIIIii[IIII llllllllil

: 0" OFF CNTR / \ :

=-- IN PLANE /%, ./ _ -_-

58.95c N,,:, ,,,,I,, i ,,,I

'"'"l"'"gh "OFF"CNTR'"' i"l"'"'"'l'"' '""l'"'""il'""'"'l'"'"'" I"'"'"'

25.20rr %_ . IN pLANE .. ' '_ ' -;

100, Ou

19.55u

142.1 160.0 180,0 200.0 220.0 240.0

Frequency (Hz)

.._ Figure 63. Channel specimen frequency response plot ti)r radial excitation; 142-240 .I.Iz
I, 538_

100, Ou

5•927u

0

0

----- 730.4u

_ 10.00u

I 854u

I I III I I I I I I I I I I I I I I I I I I I I I I I III I I I I I I_.._I I I I I I I I i I I I I I f'l I I I I I I I i I I I I I I I I

"r'-'---'_x,,._. ,. _ 5.5" OFF CMTR •-;

• "_k-_/_ / OUT oT PLANE ..-

I I III i I II I Ji If I III I t II I J I I I I I I Ifl II ill I I i t If f I I I'i'l!l if li J I I If I i,II f f

1

270.0 290,0 - ._.300,0 310.0 320,0 340,0

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I'l I I ,,,,,i,;,,,,,,,, ;,,,,,,,, ,,,,,,i,,,,,

270.0 290.0 ' 300.0 310• 0 320,0 v 340,0 _/

-,,,,,,,,,,,,,,,,, ,,,,,,,,,,

.I I !I I i'l I I il I I'II I _ I II I I II I I II II I Iil I I I III-II_ I I _ i _-

479"5uE- 9" OFF CHTR -"_'___ "z--

10" 00u i EDGEPLA.E IH v _/_H r_

1. 032u I lllillllIllllll lllIllllllllllll IIIIII]I]]I]III lllllllÁlilllllllilli

270.0 290,0 300,0 310.0 320,0 340.0

Frequency (Hz)

.._ Figure 64. Channel specimen frequency response plot for radial excitation; 270-340 ilz

oo
I l I I I I I I I I I I I I _ I I I I I I I I I I I 'i I

_ 6 OFF CMTR .-_

1:3.41

I. 000

_ J

273..9_ ' I i i i I i , , , , I , i , l , _ , , 'i''l''i''i'!. , , , i i i I ,

o _6.83s ----_------a'oo_ q. ee_ '_ I_. 11

-- la._I _- J-_ - out o_ PLANE - / ', ° 1

2A6.a_ . ',,/ . . _1

_1_ " I J I I I I I I'l 1 I J I I 1 I I I I I I I I I I I I I I I I I

o_

..6"835-----_-______________8 .000----- 9. 000-------- 10 • 1II I I I I I I' I I I I _ I.._l I I i I I I _ I I I I I I I I I I

78.75nn _ _ 9" OFF CMTR -

IN PLANE -

10.00m WEB -

J

1.953r , I _ _ , _ i , , , , I , , i i i , , _ J I , , , , , , , _ , I i

6. 835 8. 000 9. 000 10.11

Frequency (Hz)

Figure 65. Channel specimen frequency response plot fi_r lateral excitation; 6.8-10 I Iz
0

0

_21'"?"" ' ' _' ' 60F_c.TR

-15.03- -.30.00 .... 40.00 .... 50.00 60.00 80.11. - --

363"3U_1 i i1 if [,i jj i iI i ittj i ii II i jj ii i ii iljll Jl t jl jl i Ii j I, iI t Ii II J It iJ l_

.:,,,,,,,,,,,, :,,,,, ,,,,,,,, ,_,, !_,, n_9,,oF_c.T_'"'_

I. 000_ i ....... _ --

1_7._L4-_ _ i _I _ i t _ _ _ _I _ _ _ i_ _ _ _I i _ i I I jl il I I I II I II I I _ _ _ _ _ J _ _ I _ _ i _t _ J I

15.03 30.00 40.00 50.00 60.00 80.11

Frequency (Hz)

Figure 66. Channel specimen frequency response plot for lateral excitation; 15--80 itz
8

I,..i

0

°1-.i

7. 386u

589,5u'

L

I I I1 I I I I I I

I I I II I I I I I

-70.00.-- 80.00 -90.00 100.0-

I ! I I 1 I I 1 I I_1 I I I I 1 II I | I 1 I I I ! I I I "1 I ! I I I I I I I | I '! I I

'/_ . ' /._ 6" OFF CblTR' ":-I

III I I I I I t I I I I I I I i I i I I I I I ih I I I "1 I I I I I I i I I I I"

1:1.0.0 124.9--

6.361m

1.000m

516.0u

I I 'I I I I I I I I I I I i I I I I I _I I i I I I I I I I I-_- I I I I I I I I I I I 'I i I I I I

_k 0" OFF CMTR

I I 1 I I I I I I I I 1 I I 1 I I I I I I I I I 1 I I I I I I I I

27.09_

693.4u

-70.00-- -80,00"

._''''''''l'''''' ''l _r_l _l''' I' '''''''' 19',,' 0;I ;'cMTIRI''''J

=. / _f \ /X,. /'_ Ibl PLRblE _]

"-i i i i i i i i , I , i i , i i i i i I i i i t t t t i _ i I I i i H I I l I I I-(-I_i !

70.00 80.00 90.00 100.0 110.0 124,9

Frequency (Hz)

Figure 67. Channel specimen frequency response plot fi)r lateral excitation; 70-125 i it
8

o

¢1

o

0.,

,.,,_

3,580_

79.91u

4.544n

1.000m

312.1u

5. 492rr

66.29u

Frequency (Hz)

Figure 68. Channel specimen frequency response plot for lateral excitation; 140-219 i Iz
J

L,q_

o

0

0

_lll-l'll"ll" tlll[lllr[Ittllllll[Ittllllll ill IIli II1 II]lttilllll ItillllttJll|lltlll] IIIlltl; iJ 2. 636r_ =r. .-

100.0u

35.75u'ii,i,, i,k,l,ii,,'lli,tJllIIl,,i,',,'ll,,,tlillillillltlllkll,, ll,,l,,it_f,i,lllli',i,il,,i,,i,tt

--230. 0 250.0--------.--._270. 0._.--.--_290. 0..._._ 310. 0 329. B

2 i

46 13u g

• '__[__[_[______[_ II1[11 1111111

230.0 250.0 270.0 290.0 310.0 329.8- '_

p-- , •

1.302rr-__/_`_._9_.'__j_'_'''_'_H_H_''__'__''_rH_H_'_H_H_H'''_ i

100,0_ " ,_ _

|!11111111111111 III III IIIII 111111111 I111 I11111111111111111111111111111111111 II11111 IIIII1111111

230.0 250.0 270.0 290.0 310.0 329.8

Frequency (Hz)

Figure 69. Channel specimen frequency response plot for lateral excitation; 230-330 IIz
Mode Shapes

This scction presents the analytical mode shapes superimposed on tile undcformed

fiame. The modes which were presented in the text are not repeated here. The experimentally located nodes are represented by dots in the figure, the undeformed frame is

indicated by the solid line, and the mode shape is the broken line. Two free-free modes

are presentcd in the first figure. The remaining modes are for the clamped-clamped case.

The analytical mode shape was calculated by the finite element program discussed in

Chapter 5. Two plots are presented in each figure, one for the I-specimen and one for

the channel. The plots for the clamped-clamped modes are presented from top and front

views. The I-specimen mode shapes consist of pure radial or pure out-of-plane motion.

The channel modes are coupled and generally show motion in both views. The tigures

are presented without further cormnent.

Vibrational Test Dala 184
i

01

0

I w

UNDEPORM_

--- _C_ESI'IAPE

..oNODELOCATI_

(A) T_O NODE MODE

M

°•.NOOE LC_A'rION

(B) THREENOI)EMODE

l

00

Figure 70. Two and three noded in-plane, free-free mode shapes: Correlation between experimental

data and continuum solution for the free-free in-plane modes of the I-specimen.

Vibrational Test Data 185
0

_ - UODE8HA_

, • • NOOELOCATION PppFORCELOGATiON

(A) I-SP£CIM_

O,

0 eO

_ --- M_ESHAPE

• ,o NK_LOC_TION p,,FORCELOCAT'ION

(B) _ _Ec"mEN

Figure 71. Three noded in-plane, clamped-clamped modes for both specimens: The mode shapes of the

three noded in-plane modes for both specimens are presented schematically.

Vibrational Test Data 186
0

0 8O

-- _ --- I_lOOE8t-IAPE

• • • NCOE LOCATION Pp •FCI:ICE LOCATION

(A) I-SPECImEN

80

• ,, NGI_LOGATION p,PI_K_L.O_QH

(B) C]_EL SPECIMEN

Figure 72. Four noded in-plane, clamped-clamped modes for both specimens: The mode shapes of the

four noded in-plane modes for both specimens are presented schematically.

Vibrational Test Data 187
%

i,i \)

O"

• • • _ LOCATION p• PFORCELOCA_ON

(A) I- SPECIMEN

Figure 73. Seven noded in-plane, clamped-clamped modes for the l-specimen: The mode shape of the

seven noded in-plane mode for the l-specimen is presented schematically.

Vibrational Test Data 188
S4_,

--_'B'_°_'_° _1 _'_'_'_'_'_---_

0

0 !)0

_ --- MODE SHAPE

• • • _ LOCATION , ,.• FCflCE LOCATION

(A) I-SPECI_I_

0

0 80

_ --- MODE SHAPE

• .. NODE LOCATI_ ,,,FCRGELOGATION

(B) _E]. SPEO_EN

Figure 74. Two noded out-of-plane, clamped-clamped modes for both specimens: The mode shapes

of the two noded out-of-plane modes for both specimens are presented schematically.

Vibrational Test Data 189
$5,

O.

0

• • • _ LOCATION p p• FORCE LOCATION

(A) I-SPECI_

I#_4

O'

0

• °. N_OELOCATION _"FCRCIEI.OCAT_

Figure 75. Three noded out-of-plane, clamped-clamped modes for both specimens: The mode shapes

of the three noded out-of-plane modes for both specimens are presented schematically.

Vibrational Test Data 190
0'

i/" "--°--'_" ".ib I _ j,_,,a °--°_" "%,.

0o

tN::EFOR&ED --- _X]E SSAPE

•, • NODE LOCATION •,, FORCE LOCATION

(A) 1-spEcm_"

o

o 0O

•o, NODELOCATION ,,'FORCE LOC_TION

(B) CHm'_NELSPECIMEN

Figure 76. Four noded out-of-plane, clamped-clamped modes for both specimens: The mode shapes

of the four noded out-of-plane modes for both specimens are presented schematically.

Vibrational Test Data 191
SS

FS_

0

0 8O

_ --- MOI_,SHAPE

o.- NG_LOCATION p*PFORCELOCATION

CA) I-SPEC_tEN

Figure 77. Seven noded out-of-plane, clamped-<iamped mode for the /-specimen: The mode shape of

the seven noded out-of-plane mode for the l-specimen is presented schematically.

Vibrational Test Data 192
Appendix B

Static Test Data

This appendix contains some information on the static tests which was not covered

in the text. The first section presents the reduction scheme was used to calculate the

lateral and radial displacements at the centroid of the cross section at midspan. The

second section presents a sample of the reduced data from the in-plane and out-of-plane

tests.

Static Test Data Reduction

The data reduction scheme for the static tests used three measured displacements

on the load fixture to calculate the twist, _b_, of the cross section, the radial displacement,

W, and the lateral displacement, V, at the centroid of the test specimen. In general, three

measurements are sutlicient to define the location of a body moving within a plane;

hence, it is assumed that tile midspan cross section does not deform circumferentially.

We assume that the load fixture and the test specfinen deform as a rigid body, thus,

the load fixture and test specimen can be represented by a rectangular block. A generic

rcctangular body is shown in Fig. 78. A Cartesian coordinate system is placed at one

corner of the rectangular block with positive axis coinciding with the sides of the block.

The Cartesian coordinate system is consistent with the system used in the text. With

respect to the test specimen the z-axis is directed radially toward the center of the curved

frame and the y-axis is perpendicular to the frame. The distances d_, d2, and d._, locate

the three dial indicators with respect to the origin of the coordinate system. If the distance r_, and the angle, 0_, locate the ccntroid C of the cross section with respect to point

O on the load fixture, then the initial centroidal coordinates can be obtained from

Yc = rc COS 0 c

Zc = rc sin0 c (B.1)

Since the block is a rigid body, the location of the centroid can always be located

from the point O on the block. Thus, given the deformed coordinates of O, it is possible

to calculate the deformed coordinates of the centroid C. Tile deformed state of the block

is shown in Fig. 79. The deformed centroidal coordinates can be calculated from

Ya = Yo + rc cos0 c + q5x (B.2)

z c = zo + r c sin0 c + 4_x

in which yo and Zo are the deformed coordinates of point O, and _b, is the twist.

Static Test Data 193
0

i[¢

C

Y

Z

N

_O Figure 78. Schematic of rectangular block, coordinate system and dial indicators: The origin of the cartesian coordinate system coincides with point

O when the block is in the undeformed position. "Vile point C represents the centroid of the test specimen.
Z¢

d I

i I

p [ I

I I

Figure 79. Schematic of deformed cross section Since the specimen does not move with respect to the load fixture, the deformed centroidal coordinates can be calculated with respect to the detbrmed load fixture.
The centroidal displacements V and W are given by

V = Yc " Yc

, (B.3)

W = z c - z c

Thus, the unknowns necessary to calculate the centroidal displacements are

gSx, Yo, and Zo. The dial indicators are labelled 1, 2, and 3. The measurements from each

dial indicator will be referred to as in,, m2, and m3 , respectively. The points O, P, and

Q lie along line 11. The equation for line 11 can be calculated from the coordinates of

points B and C which are (d_,mj) and (d2 + dj,m2), respectively. "l'he twist, qSx; is the

slope of line 11. The equations for _x and line 11are

(m2-m 1)

d2 (B.4)

ll: z-m 2 = q5x[y-(d 1 + d2) ]

Since point O lies on line 11,the coordinate yo and Zomust satisfy Eqn. (B.4). Points

O and R define a second line, 12; which is perpendicular to line 1_, thus the slope of line

12is -1/q_,. A second equation for line 12and the coordinates Yo and Zo must satisfy this

equation as well.

1

12: z- d 3 = - _0---7"(y- m3) (B.5)

Substituting yo and Zo into Eqn. (I/.5), solving for yo, and substituting that expression

into the equation for line lI yields the following expressions for yo and Zo.

zo = {m 2 + 4_xlqS×d3 + m 3 - (dl + d2)]}

(B.6)

Yo = qSx(d3" Zo) + m3

The expressions for yo and zo can be used in conjunction with Eqs. (B.I-B.3) to obtain the lateral and radial displacements of the centroid. The special case where 4)_ is zero

corresponds to pure translation and the equations reduce to

v = m3 (B.7)

w = m 2

as expected.

Static Test Data 196
Reduced Static Data

The tables presented in this section contain a sample of the reduced data obtained

from tile reduction scheme presented in the previous section. The radial and lateral displacements and the twist at midspan for both test specimens for radial loads of twenty,

forty, and sixty pounds are presented in Tables 17, 18, and 19. The moment arm associated with each set of data is different for the two specimens. There is more data fbr the

channel specimen because the response of the channel specimen was not as consistent

as the response of the I-specimen. The flexibility influence coemcients were calculated

from this data using the least squares method discussed in Chapter 4. The radial and

lateral displacements and the twist at midspan for both specimens from the out-of-plane

tests are presented in Tables 20 and 21. For the out-of-plane tests, the moment arms

were the same for tile two specimens but the lateral loads were different. The test data

for lateral loads of 2.187 and and 1.063 pounds are presented in Table 20 for the Ispecimen and channel specimen, respectively. The test data for lateral loads of 3.25 and

2.00 pounds are presented in Table 21 for the I-specimen and the channel specimen, respectively.

Static Test Data 197
W

.,1

Table 17. Static data from in-plane tests for 20 pound radial load

l-Specimen

Moment Displacements

Arm Radial Lateral Twist

Moment

Arm

Channel Specimen

l)isplacements

Radial Lateral Twist

D W V 4,

in in in tad

X l0 -3 X 10 -3 X 10 -2

-. 1458 17.14 -27.68 -.335

-.1250 17.10 -23.44 -.279

-.1042 17.06 -16.84 -.196

-.0833 17.04 -15.60 -.140

-.0625 17.27 -12.(/4 -.098

-.0417 17.25 - 5.12 -.(128

-.0208 17.62 - 4.47 -.007

-.0104 17.36 - 1.23 ,049

0.0000 17.6 ! 1.23 .049

0.0104 17.36 4.52 .105

0.0208 17.49 5.92 .126

0.0417 17.62 9.26 .161

0.0625 17.75 11.97 .224

0.0833 17.76 15.78 ,265

0.1042 17.67 24.11 .356

0.1250 17.44 27.36 .412

0.1458 18.22 32.23 .440

D

in

w v 4,

in in tad

X I0 -3 X 10- 3 X 10- _

-.0625

-.0590

-.0555

-.0521

-.(1486

-.0451

-.(1417

-.0347

-.0243

-.0208

-.0139

-.0104

-.0069

-.0035

0.0000

28.84

28.99

29.64

29.58

29,78

29.97

30.27

30.66

31.06

31,11

31.72

32,12

32.32

32,83

33.14

-23.92

-23.01

-21.60

- 19.40

- 18.69

- 16.74

-15.17

-12.25

- 9.09

- 7.02

- 6.41

- 4.25

- 0.04

- 0.26

- 0.27

7.51

7.47

8.17

11.49

14.20

12,90

16.30

20.25

22,74

24.19

23.49

0,0035

0,0069

0,0104

0.0139

0,02O8

0.0243

0.0347

0.0382

0,0417

0.0451

0.0486

33.34

33.55

33.75

34.31

34.77

34.72

35.11

35.32

36.31

36.02

35.81

-.370

-.350

-.329

-.319

-.308

-.298

-.257

-.236

-.216

-.176

-.144

-.123

-.113

-.062

-.021

-.010

0.000

0.010

0.051

0.062

0.(172

0.154

0.165

0.226

0.236

0.226
.-4

¢1}

Tablc 18. Static data from in-planc tests for 40 pound radial load

l-Spccinlcn

Moment i)isplaccmcnts

Arm Radial Lateral Twist

D w v 4,

in in in rad

x 10-3 x 10-3 x 10--2

Molncll |

Arm

D

in

Channel Specimen

i)isl_laccnlents

Radial Lateral Twist

W V 4'

in in tad

x 10- _ x 10-3 x 10- 2

-. 1458 33.25 -56.40 -.6840

-.1250 33.13 -49.39 -.5870

-.1042 33.35 -36.43 -.4260

-.0833 33.54 -32.09 -.3420

-.0625 33.45 -22.20 -.2160

-.0417 33.41 -12.18 -.1190

-.0208 33.62 - 5.03 -.0000

-.0104 33.62 .99 0.0489

0.0000 33.86 !.06 0.0908

0.0104 33.62 7.91 O. 1746

0.0208 33.63 11.90 0.2304

0.0417 34.02 19.10 0.2933

0.0625 34.07 27.26 0.4190

0.0833 34.12 33.61 0.5028

O. 1042 33.99 46.65 0.6564

0.1250 33.82 51.45 0.7542

0.1458 34.83 62.77 0.8450

-.0625 56.94 -55.77 -.7500

-.0590 57.45 -51.21 -.7090

-.0555 57.69 -49.46 -.7090

-.0521 58.72 -43.35 -.6270

-.0486 59.30 -40.24 -.5960

-.0451 60.07 -36.18 -.5550

-.0417 60.73 -31.22 -.4930

-.0382 61.30 -24.87 -.4630

-.0347 62.04 -23.86 -.4110

-.0243 61.99 -22.40 -.4010

-.0208 62.72 -17.40 -.3505

-.0139 64.32 -12.02 -.2160

-.0069 64.90 - 2.17 -.1850

-.0035 66.07 - 2.00 -.1130

0.0000 66.58 6.26 -.0620

0.0035 67.23 11.16 .0103

0.0069 67.30 14.72 .0000

0.0104 67.98 19.13 .0720

0.0139 68.92 24.79 .1439

0.0208 69.42 31.01 .2056

0.0313 70.91 37.93 .3289

0.0347 70.63 41.63 .3392

0.0382 71.74 46.19 .3803

0.0417 73.12 50.26 .4729

0.0451 73.86 57.70 .4831

0.0486 74.00 59.81 .5148
,...1

1".,4

Table 19. Static data from in-planc tests for 60 pound radial h_ad

l-Specimen

Moment l)isplacemcnts ]Moment

Channel Sl_ecimcn

I)isplaccmcnts

Arm Radial Lateral Twist

D w v 4,

in in in tad

x I0 -j x I0 -_ x I0 -z

Arm Radial Lateral Twist

D W V 4,

in in in tad

x 10-_ x I0 -_ x I0 -z

-. 1458 49.93 -.0908 - 1.0750

-. 1250 49.60 -.0764 - .9080

-.1042 49.40 -.0595 - .6910

-.0833 49.26 -.0529 - .5660

-.0625 49.43 -34.93 - .3630

-.0417 49.57 -22.52 .1960

-.0208 49.88 - 9.40 - .0210

-.0104 49.99 - 5.03 .0559

0.0000 50.11 0.98 .1047

0.0104 50.(/0 11.33 .2374

0.0208 50.90 19.38 .3282

0.0417 50.44 26.22 .4120

0.0625 50.21 42.93 .6285

0.0833 50.34 53.82 .7542

O. 1042 50.61 70.26 .97?7

0.1250 49.96 82.14 1.1103

0.1458 50.63 96.20 1.2573

-.0625 84.13 - 102.00 - !.203

-.0590 84.82 - 95.70 -1.162

-.0555 85.34 - 91.11 -1.131

-.0521 86.60 - 80.93 - 1.018

-.0486 88.01 - 72.14 -0.9250

-.0451 89.14 - 67.45 -0.8630

-.0417 89.89 - 59.86 -0.7_;10

-.0382 91.59 - 50.57 -0.6890

-.0347 92.09 - 45.53 -0.6480

-.0278 92.84 - 38.99 -0.6064

-.0208 93.73 - 30.56 -0.4935

-.0139 95.58 - 19.17 -0.3700

-.0104 96.57 - 17.43 -0.3190

-.0069 97.45 8.79 -0.2470

-.0035 98.41 3.85 -0. i 850

0.0000 99.37 6.83 -0.0720

0.0035 100.70 17.86 0.0206

0.0069 10i .39 24.0(I 0.0925

0.0104 102.46 29.74 O. 1953

O.0139 103.75 39.42 O.3084

0.02(18 105.17 53.46 0.40(19

0.0313 107.57 65.69 O. 5(137

0.0382 108.40 77.13 0.5654

0.0417 110.50 86.76 0.6887

0.0451 111.41 99.90 0.8121

0.0486 111.68 98.45 0.8018
,-]

Table20. Static data from out-of-plane tests

;¢

I-Specimen, Q = 2.187 Ibs

Molncll|

Arm

D

in

Displacements

Radial Lateral Twist

w v

in in rad

x 10- :_ x 10 -3 X 10-I

MolilCl| |

Arm

1)

in

Channel Specimen, Q = 1.063 Ibs

I)ispl:wcments

Radial Lateral Twist

w v 4,

in in rad

X 10 -3 X 10-3 x I0 -_

-0.6850 2.49 232.30 O.1746

-0.6294 2.29 233.24 O. 1760

-0.5739 2.08 235.24 O. 1760

-0.5183 2.36 234.93 0.1774

-0.4628 2.28 236.91 0.1781

-0.4072 2.37 237.79 O. 1809

-0.3517 2.23 238.07 0.1802

-0.2961 2.44 239.74 O.1823

-0.2406 2.37 240.15 O. 1844

-0.1850 2.49 240.68 0.1837

-0.1294 2.61 243.57 0.1865

-0.0739 2.64 243.01 O. 1878

-0.0185 2.58 241.82 0.1865

0.0372 2.68 243.45 O. 1893

0.0928 2.43 247.37 0.1913

0.1483 2.78 248.75 0.1941

-0.6850 7.94 242.21 O. 1145

-0.6294 8.11 244.68 O. 1152

-0.5739 8.01 244.90 O. 1159

-0.5183 8.19 244.81 0.1180

-0.4628 8.13 246.47 O. 1201

-0.4072 8.29 247.94 O.1208

-0.3517 8.74 250.09 O. 1229

-0.2961 8.44 248.41 O. 1215

-0.2406 8.64 250.06 O.1237

-0.1850 8.56 252.79 O. 1244

-0.1294 8.73 254.50 O. 1250

-0.0739 8.77 255.19 0.1264

-0.0185 8.79 256.69 O. 1264

0.0372 9. I0 258.63 O. 1278

0.0928 9.17 260.82 O. 1292

O. 1483 9.17 261.32 O. 1292

m
oq

,.'4

Table 2 I. Static data from out-of-plane tests

l-Specimen, Q = 3.25 Ibs

Moment

Arm

D

in

l)isplacements

Radial Lateral Twist

w V 4,

in in rad

X 10 -3 X I0 -_ x I0 -I

Mollll_llt

Arm

D

in

Channel Specimen, Q = 2.000 Ibs

l)isplaccmcnts

Radial l.atcral Twist

w v 4,

in in rad

X 10 -3 X I0 -_ x 10-'

-0.6850 4.31 345.59

-0.6294 4.16 347.54

-0.5739 4.11 349.51

-0.5183 4.00 348.99

-0.4628 4.32 353.26

-0.4072 4.15 354.22

-0.3517 4.15 356.13

-0.2961 4.57 358.52

-0.2406 4.47 358.24

-0.1850 4.37 361.94

-0.1294 4.84 362.35

-0.0739 4.90 362.55

-0.0185 4.70 362.50

0.0372 5. i 8 365.(17

0.0928 4.99 368.50

O. 1483 4.87 371.20

0.2612

0.2626

0.2633

0.2640

0.2696

0.2710

0.2730

0.2758

0.2765

0.2779

0.2800

0.2814

0.2828

0.2870

0.2891

0.2905

-0.6850 13.94 434.27 0.2067

-0.6294 14.02 438.02 0.2067

-0.5739 13.90 436.99 0.2074

-0.5183 14.15 438.40 0.2095

-0.4628 14.30 443.76 0.2130

-0.4072 14.20 443.48 0.2137

-0.3517 14.54 446.08 0.2 ! 72

-0.2961 14.71 446.80 0.2 i 79

-0.2406 14.78 447.24 0.2193

-0.1850 14.98 450.37 0.2221

-0.1294 15.05 453.37 0.222 i

-0.0739 15.21 455.25 0.2249

-0.0185 15.36 458.69 0.2263

O. 0372 15, 51 459.33 0.229 I

0.0928 15.58 462.33 0.229 I

O. 1483 15.69 464.27 0.2305

t_
Appendix C

Cross-Sectional Properties

The modulus-weighted and geometric cross-sectional properties of the two specimens are presented in Table 22. The geometric section properties were obtained using a

computer program listed in ReF. 33. The modulus-weighted section properties were obtained by modifying this computer program to evaluate the expressions for the

modulus-weighted section properties presented in Chapter 5. The modulus-weighted

section properties are normalized by the effective engineering moduli determined by the

coupon tests discussed in Chapter 2. The mass density of the material measured to be

1465. kilograms per cubic meter.

Table 1. Cross-sectional properties of both test specimens

Section I-specimen

Property M-W'

Channel Specimen

Geometric _ M-W' Geometric b

A m2 0.3658 x 10-3 c 0.3310 x 10 -3 0.2323 x 10-3 c 0.2102 x 10 -3

Ay m2 0.3658 x 10-3 d 0.3310 X 10-3 0.2323 X 10-3 a 0.2102 X 10 -3

A, m 2 0.3658 x 10-3 d 0.3310 x 10-3 0.2323 x 10 -3 d 0.2102 x 10-3

I_ m" 0.5339 x 10-7 ¢ 0.4813 x 10-7 0.3202 x 10 -7 c 0.2888 x 10-7

I_, m 4 1.6705 x 10-7 c 1.5114 x 10 -7 0.4844 x 10 -7 c 0.4382 x 10-7

I. m 4 0.0000 x 10-7 c 0.000 x 10-7 -.1070 x 10-7 c -.9685 x 10 -s

J m 4 0.8842 x 10-9 d 0.7825 x 10 -9 0.5026 x 10-9 d 0.4378 x 10 -9

S,_ m s 0.9671 x 10-7 _ 0.8755 x 10 -7 0.1089 x 10-7 _ 0.9860 x 10 -8

l_,y m s -.3217 x 10-_s c -.4539 x 10-_s -.6650 x 10-9 c -.6023 x 10-9

I_,, rn 5 0.8114 x 10-9 ¢ 0.7343 x 10 -9 0.4301 x 10-9 ¢ 0.3829 x 10-9

I_,,_ m 6 0.3558 x 10-l°' 0.3213 x 10-l° 0.1688 x 10-I° c 0.1526 x 10-_°

Modulus-Weighted section properties from Chapter 5

Geometric section properties from Ref. 33

Normalized by E= 5.075 x 10 _° Nt/m 2

Normalized by G= 1.953 x I0 l° Nt/m 2

Cross-Sectional Properties 203
NAS 

I. Report No. 2. Government Accession No.

NASA CR-4287

4. Title and Subtitle

Static and Free-Vibrational Response of

Semi-Circular Graphite-Epoxy Frames With

Thin-Walled Open Sections

7. Authorts)

J. Scott Collins

Eric R. Johnson

and i

Report Documentation Page

9. Pedormlng Organization Name and Addre_

Virginia Polytechnic Institute and State

University

Dept. of Engineering Science and Mechanics

106 Norris Hall Blacksburq, VA 24061

3. Rec=p_ent'sCatalog No.

5. Report Date

March 1990

6. Performing Organization Code

8. Performing Organization Repo_ No.

VPI-E-89-19

12. Sponsoring Agency Name and Addre_

NASA Langley Research Center

Hampton, VA 23665-5225

10. Work Unit No.

505-63-01-11

11. Contract or Grant No.

NAGI-343

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

15. Supple_nta_ Notes

Langley Technical Monitor: Huey D. Carden

16. Abstract

Experiments were conducted to measure the three-dimensional

static and free vibrational response of two graphite-epoxy,

thin-walled, open section frames. The frames are semi-circular with

a radius of three feet, and one specimen has an I cross section and

the other has a channel cross section. The flexibility influence

coefficients were measured in static tests for loads applied at

midspan with the ends of the specimens clamped. Natural frequencies

and modes were determined from vibrational tests for free and

clamped end conditions. The experimental data is used to evaluate

a new finite elements which was developed specifically for the

analysis of curved, thin-walled structures. The formulation of the

element is based on a Vlasov-type, thin-walled, curved beam theory.

17. Key Words (Suggestodby Author(s))

Crashworthiness

Composite Materials

Graphite-Epoxy

Impact Loading

Nonlinear Response

19. Secur_WCla_ff. (of thts report) Unclassified

Beam Theory

Vibration

Response

18. Distribution Statement

Unclassified - Unlimited

Subject Category: 39

20. Securiw Cla_ff. (of this page)

Unclassified

21, No. of pages

2O8

NASA FORM 1626 OCT86

For sale by the National Technical Information Se_'ice, Spring-field, Virginia 22161-2171

22. Prtce

A10

NASA-Langley, 1990
÷

_t

If Uitdeliverable (Section 158

Postal Manual) Do Not Return

o